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Nonlinear dynamical modeling of single neurons and its application to analysis of long-term potentiation (LTP)

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Nonlinear dynamical modeling of single neurons and its application to analysis of long-term potentiation (LTP)

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Content Copyright 2011 Ude Lu

NONLINEAR DYNAMICAL MODELING OF SINGLE NEURONS AND ITS
APPLIACTION TO ANALYSIS OF LONG-TERM POTENTIATION (LTP)

by

Ude Lu
________________________________________________________________________

A Dissertation Presented to the
FACULY OF THE USC GRADUATE SCHOOL
UNIVERSITY OF SOUTHERN CALIFORNIA
In Partial Fulfillment of the
Requirements of the Degree
DOCTOR OF PHILOSOPHY
(BIOMEDICAL ENGINEERING)

August 2011

ii

DEDICATION

To my dearest wife, Fang-Ting,
and to my supportive parents,
without them this thesis is impossible.

iii

ACKNOWLEDGMENTS

I am extremely grateful to the members of my dissertation committee, Dr.
Theodore Berger, Dr. Dong Song, Dr. Michel Baudry, Dr. David Z. D’Argenio, and Dr.
Vasilis Marmarelis, the principle contributors to this work, for their guidance and
instruction. I specially thank my advisors, Dr. Theodore Berger and Dr. Dong Song for
providing tremendous amount of time and support throughout the entire project. I also
thank Dr. John Walsh for being on my qualifying exam committee.
This work is also indebted to the contributions of other graduate students and
post-doctoral fellows in the Berger Lab. Dr. Min-Chi Hsiao taught me the basics of
performing electrophysiological experiments. Rosa Chan provided many discussions on
programming. Shane Roach collaborated with me on parts of the experimental works. I
am equally grateful to other lab members, Dr. Jean-Marie Bouteiller, Sushmita Allam,
Penning Yu, and Viviane Ghaderi for their companionship, encouragement, and support.

iv

TABLE OF CONTENTS

DEDICATION……………………………………………………………………………ii

ACKNOWLEDGMENTS………………………………………………………………iii

LIST OF FIGURES………………………………………………………………………vi

ABSTRACT……………………………………………………………………………viii

CHAPTER 1: BACKGROUND…………………………………………………………1
1.1. Research purpose…………………………………………………………………1
1.2. Single neuron modeling methodologies…………………………………………3
1.2.1. Compartmental modeling…………………………………………………3
1.2.1.a. Hodgkin-Huxley model………………………………………………3
1.2.1.b. Cable theory…………………………………………………………..6
1.2.2. Integrate and fire model…………………………………………………8
1.2.3. Input-output modeling: Volterra series…………………………………10
1.2.4. Discussion………………………………………………………………12
1.3. Hippocampus CA1 pyramidal neurons and its short- and long-term
plasticity………………………………………………………………………14
1.3.1. Functions of the hippocampus…………………………………………14
1.3.2. General introduction to CA1 pyramidal neurons………………………15
1.3.3. Short-term plasticity……………………………………………………17
1.3.4. Long-term potentiation…………………………………………………19
1.4. Whole-cell patch clamp………………………………………………………23
1.4.1. Introduction………………………………………………………………23
1.4.2. Circuits…………………………………………………………………24
1.5. References………………………………………………………………………26

CHAPTER 2: HIGH-ORDER DYNAMICAL MODLEING OF SINGLE NEURONS
WITH CONSTANT THRESHOLD……………………………………………………31
2.1. Introduction……………………………………………………………………31
2.2. Experiment procedures…………………………………………………………37
2.3. Model structure and its mathematical expressions…………………………39
2.4. Model estimation procedures…………………………………………………42
2.4.1. Parameter optimizations and estimations………………………………43
2.4.1.a. Optimization of Laguerre parameters………………………………44
2.4.1.b. Optimization of the total number of Laguerre basis functions……46
2.4.1.c. Estimation of Laguerre expansion coefficients……………………46
2.4.1.d. Optimization of constant threshold…………………………………47
2.4.2. Model interpretations……………………………………………………48
2.4.2.a. Reconstructions of Volterra kernels…………………………………48

v

2.4.2.b. Calculations of response functions…………………………………49
2.4.2.c. K
1
model……………………………………………………………50
2.4.2.d. K
2
model……………………………………………………………51
2.4.2.e. K
3
model……………………………………………………………53
2.4.3. Evaluations and quantifications…………………………………………55
2.4.4. Out-of-sample predictions………………………………………………56
2.4.4.a. Representative prediction clips……………………………………56
2.4.4.b. PSP predictions……………………………………………………57
2.4.4.c. Spike predictions……………………………………………………59
2.5. Discussion………………………………………………………………………61
2.6. References………………………………………………………………………65

CHAPTER 3: DYNAMIC THRESHOLD MODEL AND ITS INTEGRATION WITH
THE SINGLE NEURON MODEL………………………………………………………68
3.1. Introduction……………………………………………………………………68
3.2. Measuring threshold……………………………………………………………72
3.2.1. Action potential (AP) turning point measurement………………………72
3.2.2. The constant offset between AP turning point and threshold……………76
3.3. Modeling threshold dynamics…………………………………………………78
3.3.1. Estimation of the dynamic threshold model……………………………78
3.3.2. Reconstructions of Volterra kernels……………………………………80
3.3.3. Response functions………………………………………………………81
3.4. Integration of the dynamic threshold model with the single neuron model……84
3.4.1. Spike prediction accuracy improvements………………………………84
3.5. Discussion………………………………………………………………………90
3.6. References………………………………………………………………………93

CHAPTER 4: NONLINEAR DYNAMICAL MODELING OF LONG-TERM
SYNAPTIC PLASTICITY………………………………………………………………96
4.1. Introduction……………………………………………………………………96
4.2. Materials and methods…………………………………………………………100
4.2.1. Electrophysiology………………………………………………………100
4.2.2. Estimating vesicle release strengths from the recorded random-internal
EPSCs…………………………………………………………………101
4.2.3. Volterra modeling of pre- and post-synaptic mechanisms……………102
4.2.4. Reconstructions of Volterra kernels and response functions…………104
4.2.5. The linearity of AMPA
RC
-mediated current dynamics………………104
4.3. Results and discussion………………………………………………………107
4.4. References……………………………………………………………………114

ALPHABETIZED REFERENCES……………………………………………………117

vi

LIST OF FIGURES

Fig. 1-1. Hodgkin-Huxley model for squid giant axons…………………………………4
Fig. 1-2. Cable theory……………………………………………………………………7
Fig. 1-3. The integrated-and-fire model…………………………………………………9
Fig. 1-4. (a) Hippocampus neuroanatomy. (b) CA1 pyramidal neuron morphology……15
Fig. 1-5. Four gigaseal recording methods………………………………………………23
Fig. 1-6. Electronic scheme of current to voltage converter for whole-cell recording…25
Fig. 2-1. Experimental paradigm………………………………………………………33
Fig. 2-2. Single neuron model structure…………………………………………………35
Fig. 2-3. Optimizations of  and L………………………………………………………45
Fig. 2-4. Evaluations of model prediction performance with normalized mean square
error (NMSE) and spike prediction error rate (SPER)…………………………………48

Fig. 2-5. The response functions of a K
1
model…………………………………………51
Fig. 2-6. The response functions of a K
2
model…………………………………………52
Fig. 2-7. The response functions of a K
3
model…………………………………………54
Fig. 2-8. A representative clip of recorded data (the upper panel) and the corresponding
predicted data (the lower panel) with an optimal threshold (dashed line)………………57

Fig. 2-9. Distributions of NMSEs and NMSE improvements with increasing model
orders……………………………………………………………………………………58

Fig. 2-10. Distributions of SPERs and SPER improvements with increasing model
orders...…………………………………………………………………………………60

Fig. 3-1. The dynamic threshold model. The model takes AP firing history as input and
threshold value as output………………………………………………………………71

Fig. 3-2. The third-order derivative analysis of AP turning point………………………73

vii

Fig. 3-3. AP turning point and threshold analyses………………………………………75
Fig. 3-4. Estimation of dynamic threshold model………………………………………80
Fig. 3-5. Threshold dynamics response functions………………………………………83
Fig. 3-6. Model structure of the single neuron model with the constant threshold and
the dynamic threshold……………………………………………………………………85

Fig. 3-7. Representative sample clips of spike predictions made with constant
threshold and dynamic threshold………………………………………………………87

Fig. 3-8. ROC curve analysis……………………………………………………………89
Fig. 4-1. Model structure of the two-stage cascade model………………………………98
Fig. 4-2. recordings of EPSCs and EPSPs evoked by the same pre-synaptic RIT
stimulation pattern……………………………………………………………………100

Fig. 4-3. The transformation from pre-synaptic spikes to EPSCs involves pre-synaptic
mechanisms and AMPA
RC
-mediated current dynamics………………………………101

Fig. 4-4. The reconstructed EPSCs produced by convolving ys(t) and hc(t).…………105

Fig. 4-5. A representative clip of EPSC recordings before and after LTP……………107
Fig. 4-6. A representative clip of EPSP recordings before and after LTP……………107
Fig. 4-7. Response functions of presynaptic model before and after LTP……………108
Fig. 4-8. Response function of postsynaptic model before and after LTP……………110
Fig. 4-9. Presynaptic and postsynaptic nonlinearity changes…………………………112

viii

ABSTRACT

Neuron spike-train to spike-train temporal transformation is very important to the
functions of neurons. Neurons receive presynaptic (input) spike-trains and transform
them into postsynaptic (output) spike-trains. This input-output transformation is a highly
nonlinear dynamic process which depends on complex nonlinear physiological processes.
Mathematically capturing and quantifying neuron spike-train to spike-train
transformation are important to understand the information processing done by neurons.

Compartmental modeling methodology is to simulate and interpret detail neuron
physiological mechanisms/processes. The Hodgkin-Huxley model is the most prominent
example in this category. However, model structure/parameter of compartmental
modeling is specific to the targeted neuron (or type of neurons) and not applicable to the
others, and the modeling result is vulnerable to biased or incomplete knowledge. Hence,
the number of open parameters is often large, making it computationally inefficient.
Integrate-and-fire neuron model is a computationally efficient methodology that received
a lot of attention in the past two decades. It is perfect for large-scale simulation, and
provides qualitative neuron characterization. However, it is over simplified and provides
no or little mechanistic implications or quantifications. Lastly, input-output modeling
methodology, which is applied in this study is another major approach to characterize
neuron spike-train transformation. Input-output models are data-driven. This leads to an

ix

important property that it avoids modeling errors due to biased or incomplete knowledge.
The number of open parameters is limited, making the model relatively computationally
efficient. In other words, input-output model provides is well balanced between the
common modeling dilemma: accuracy and efficiency.

In my study, the purpose is to build a single neuron model that 1) captures both
sub- and supra-threshold dynamics based on neuron intracellular activity, 2) is
sufficiently general to be applied to all spike-input, spike-output neurons, 3) is
computationally efficient.

A nonlinear dynamical single neuron model was developed using Volterra kernels
based on patch-clamp recordings. There were two phases in developing this model. In the
first phase, a single neuron model with constant threshold was developed. It consists: 1)
feedforward kernels (up to third-order) which transform presynaptic spikes into
postsynaptic potentials (PSPs), 2) a constant threshold which represents the spike
generation process, and 3) a feedback kernel (first-order) which describes spike-triggered
after-potentials. The model was applied to CA1 pyramidal cells as they were electrically
stimulated with broadband impulse trains through the Schaffer collaterals. This
synaptically driven broadband intracellular activities contains a broad range of nonlinear
dynamics resulted from the interactions of underlying mechanisms. The model
performances were evaluated separately with respect to: PSP waveforms and the

x

occurrence of spikes. The average normalized mean square error (NMSE) of PSP
prediction is 14.4%. The average spike prediction error rate (SPER) is 18.8%.

In the second phase, inspired by literatures, a dynamical model was developed to
study threshold nonlinear dynamics according to the action potential (AP) firing history.
To develop the model, we measured the turning point of AP by analyzing its third-order
derivative. The AP turning point has a constant offset relationship with the threshold. In
other words, variation to the AP turning point represents the nonlinearities of threshold
dynamics. To perform accurate spike prediction, it requires an additional spike prediction
validation to optimize that offset (the linearity). This dynamic threshold model was
implemented using up to third-order Volterra kernels constrained by synaptically driven
intracellular activity described before. This threshold model was integrated into the single
neuron model to replace its original constant threshold and showed 33% SPER
improvement.

This single neuron model is a hybrid, combining both mechanistic (parametric)
and input-output (non-parametric) components. The principles of neuronal signal
generation common to all spike-input, spike-output determine the model structure. On the
other hand, the specific properties that are variable from neuron to neuron are captured
and quantified with descriptive model parameters, which are directly constrained by
intracellular recording data. This hybrid representation of both parametric and
nonparametric model components partitions data variance with respect to mechanistic

xi

sources and thus imposes physiological definitions to the model components and
facilitates the biological interpretations of the parameters.

This single neuron model was further applied to analyze long-term potentiation
(LTP) in single neurons. The purpose of this application is to separate and quantify the
pre- and post-synaptic mechanisms both before and after LTP induction. The single
neuron model is modified to be a two-stage cascade model. The first-stage represents
presynaptic mechanisms, taking presynaptic spikes as input and excitatory postsynaptic
currents (EPSCs) as output. The second-stages represents postsynaptic mechanisms,
taking EPSCs as input and excitatory postsynaptic potentials (EPSPs) as output.
Preliminary data shows that LTP intensifies the linear responses and reduces the
nonlinearities.

1

CHAPTER 1: BACKGROUND

1.1. Research purpose
Neuron spike-train to spike-train temporal transformation is very important to the
functions of neurons. Neurons receive presynaptic spike-trains (input) and transform
them into postsynaptic spike-trains (output). All physiological functions performed by
neurons, to name a few, such as learning and memory, is embedded in such spike-train
temporal transformations. These spike-train transformations are highly nonlinear
dynamical processes which are the result of the complex interactions of various
underlying physiological mechanisms. The transformations involve calcium-influx in
presynaptic terminal, presynaptic vesicle release of neurotransmitter, postsynaptic
transduction, synaptic integration, somatic integration, AP generation, AP back-
propagation into the dendritic arbors, and retrograde signaling towards the presynaptic
terminal [1-4].
Mathematically capturing and quantifying this neuron spike-train to spike-train
transformation are critical to understand the information processing done by neurons. In
my study, the purpose is to build a single neuron model that 1) captures both sub- and
supra-threshold nonlinear dynamics based on synaptically-driven broadband neuron
intracellular activity, 2) is sufficiently general to be applied to all spike-input, spike-
output neurons, 3) is computationally efficient.
Many single neuron modeling methodologies have been developed for different
purposes. The particular approach used depends largely on the purposes of the modeling
2

study. In other words, the goal of the study requires a choice among modeling
methodologies. There are three major methodologies in current neuron modeling practice:
compartmental model, integrate and fire model, and nonparametric nonlinear dynamic
model. I will briefly introduce the advantages and limitations of each of them.
3

1.2. Single neuron modeling methodologies
1.2.1. Compartmental modeling
Compartmental modeling is generally used to simulate and interpret underlying
mechanisms. Many successful compartmental models simulating different system levels
of neuron from ion channel, membrane, synapse, dendrite, and whole neuron were build.
However, the most prominent example in compartmental modeling methodology is
Hodgkin-Huxley model.

1.2.1.a. Hodgkin-Huxley model
Neuroscientists' attempt to explain the underlying mechanism of neurons in terms
of mathematical equations has never stopped. One particularly successful approach,
Hodgkin-Huxley (HH) model in 1952, explains the generation and propagation of action
potentials using simple electrical circuits and insightful differential equations [5].
Alan L. Hodgkin and Andrew Huxley (1952) [6] developed a spike generating
axon model to fit the empirical data obtained through experiments on squid giant axons,
shown in Fig. 1.1.
4

Fig. 1.1. Hodgkin-Huxley model for squid giant axons.

In this model, Hodgkin and Huxley expressed the passive properties of neuron
membranes as those of a capacitor and a resistor; and expressed the active properties
using two ion-specific voltage dependent conductance. The total current injected into the
membrane, I(t), is expressed in the following way,
() () () () ()
m
CK Na leak
It I tI tI tI t    (1.1)
where I
C
, I
K
, I
Na
, and I
Leak
are currents flowing through g
C
, g
K
, g
Na
, and g
Leak
, respectively.
From the definition of a capacitor, we know that
5

() () () () ()
mC K Na leak
du
C I t ItI tI tI t
dt
    (1.2)
where u represent the membrane voltage. The ion specific currents ( )
K
I t, ()
Na
I t , and ion
nonspecific leaking current ( )
Leak
I t are expressed as follows,
4
() ( ( ) )
K Kk
I tgnut E  (1.3)
3
() ( ( ) )
Na Na Na
I tgmhut E  (1.4)
() ( ( ) )
leak leak rest
It g ut E  (1.5)
Where
2
36 ( / )
K
gmScm  ,
2
120 ( / )
Na
gmScm  , and
2
0.3 ( / )
Leak
gmScm  represent
the maximum ion channel conductance; and 77
K
EmV   , 50
Na
EmV  , and
65
Leak
EmV  are the reversal potentials. The channel gating variables m, n, and h are
expressed as,
()(1 ) ( )
()(1 ) ( )
()(1 ) ( )
mm
nn
hh
mu m um
nu n un
hu h uh









(1.6)
where all the α and β are empirical functions to fit the data obtained from squid giant
axons expressed as,
( ) (0.1 0.01 ) /[exp(1 0.1 ) 1]
( ) 0.125exp( / 80)
n
n
uu u
uu


  

(1.7)
( ) (2.5 0.1 ) /[exp(2.5 0.1 ) 1]
( ) 4exp( /18)
m
m
uu u
uu


  

(1.8)
( ) 0.07exp( / 20)
( ) 1/[exp(3 0.1 ) 1]
h
h
uu
uu



 
(1.9)
6

The HH model is established as a classical frame work and is widely used in
compartmental modeling, even now. Upon the discovery of new ion channels over time,
more channels were added to the model and expressed in similar equation structures. The
HH model is very successful in interpreting the relationship of the membrane voltage and
ion channel dynamics. However, large number of open parameters and heavy
computational power consumption make it inapplicable for large-scale simulation, unless
very high standard of hardware requirements are met [5].

1.2.1.b. Dendrite model: cable theory
Dendrites model: cable theory
Hermann in 1905 developed the so-called the cable theory describing the relations
between voltage propagations and space distribution of dendrite. This model continues to
be a widely adopted dendrite model [7]. See Fig. 1.2.
7

Fig. 1.2. Cable theory

Considering dendrite being decomposed into short segments of cylinder with length dx. A
longitudinal current i(x) passing through the dendrite causes a voltage drop across the
longitudinal resistor R
L
according to Ohm’s law.
(, ) ( , ) (, )
L
u t x dx u tx Ri tx   (1.10)
where u(t,x) is the membrane potential at the neighboring point x+dx.
According to Kirchhoff’s law regarding the conservation of current at each node, this
leads to
(, )
(, ) (,) (, ) (,)
ext
T
ut x
it x dx i t x i t x C u t x
tR

   

(1.11)
After some derivation, the cable equations regarding membrane voltage are expressed as
follows,
8

2
2
(, ) ( , ) (, ) ( , )
ext
ut x u t x ut x i t x
tx

 

(1.12)
The cable equation regarding current is expressed as,
2
2
(, ) ( , ) (, ) ( , )
ext
it x i t x it x i t x
tx x
 
 
 
(1.13)
To intuitively describe cable theory, the longitudinal voltage and/or current at a certain
point in a dendrite decreases exponentially with distance along dendrites.
Cable theory is often combined with HH model to construct a model for a whole
neuron. This kind of neuron model has to be simplified to certain level, since it is not
impossible to build a neuron model with tens of thousands of dendrites as there are in real
biological systems.
In summary, compartmental modeling methodology is to simulate and interpret
detail neuron physiological mechanisms/processes. The Hodgkin-Huxley model is the
most prominent example in this category. However, model structure/parameter of
compartmental model is specific to the targeted neuron (or type of neurons) and not
applicable to the others. The modeling result is vulnerable to biased or incomplete
knowledge [Dong's papers]. Hence, the total number of open parameters is often large,
making it computationally inefficient.

1.2.2. Integrate and fire model
The integrate-and-fire (IF) model express neuron soma as a purely passive entity
using simple RC circuits and a constant threshold term for spike generation, as shown in
Fig. 1.3.
9

Fig. 1.3. The integrate-and-fire model.

The total current injection i(t) can be split into two components: 1) a resistor current i
R
,
and 2) a capacitor current i
C
. The relationship is expressed as,
()
()
Rc
ut du
it i i C
R dt
   (1.14)
where u(t) means the membrane voltage. The membrane voltage is then compared to a
fixed threshold to determine the spike generation.
Because it is a highly simplified expression of neuron, the IF model is very
computationally efficient and fits the need of large-scale simulations. Over time,
scientists has developed various modified IF model versions for different modeling
10

purposes [add some example]. However, IF model over simplifies neuron, and overlooks
most underlying nonlinear dynamical mechanisms of neuron.
To sum up, integrate-and-fire neuron model is a computationally efficient
methodology that received a lot of attention in the past two decades. It is perfect for
large-scale simulation, and provides qualitative neuron characterization. However, it is
over simplified and provide no or little mechanistic implications or quantifications.

1.2.3. Input-output modeling: Volterra series
Before the discussion of the input-output modeling methodology, the definition of
a linear system needs to be mentioned. Let N be a linear system, and
11
() ( ( )) ut N x t  ; and
22
() ( ( )) ut N x t  (1.15)
then,
11 2 2 1 1 2 2
(() ()) () () Ncx t c x t cu t cu t   (1.16)
The equations state that the multiplication and summation of inputs result in the same
multiplication and summation of outputs in a linear system.
Let us consider a nonlinear system case. If a system N is expressed in the
following way
12 1 2 1 2
00
(()) ( , )( )( )
TT
Nxt h xt xt d d   

(1.17)
The response to an impulse, () t  , is given by
11

12 1 2 1 2
00
111
0
(()) ( , )( )( )
(,) ( )
(, )
TT
T
Nt h t t dd
ht t d
ht t
   
  





(1.18)
If we multiply the input δ(t) by α,
12 1 2 1 2
00
2
(()) (, )( )( )
(, )
TT
Nt h t t dd
ht t





(1.19)
This means that
2
( ( )) ( ( )) Nut Nut   ; This equation does not follow the definition of a
linear system, meaning that N is a nonlinear system. In 1959, Volterra generalized the
concept of convolution to deal with nonlinear systems by replacing the single impulse
response with a series of multi-dimensional integration kernels [8, 9]. The resulting
Volterra series is given by
() 1 1 1
0
00
() ( , , ) ( ) ( )
qq q q
q
ut k x t x t d d    






  
(1.20)
where k means Volterra kernel, and q means the order of kernels. The same equation in
discrete form is:
1
() 1 1 2
00 0
() ( ) ( ) ( )
q
MM
qq
q
ut k x t x t

 

 

 
 
(1.21)
where M is a finite memory window
The Volterra series treats a nonlinear system as a black box, disregarding the
detailed mechanisms working in the system but accurately captures the input-output
transformations using various orders of kernels k
(q)
.
12

Lastly, input-output modeling methodology, which is applied in this thesis is
another major approach to characterize neuron spike-train transformation. Input-output
models are data-driven. This leads to an important property that it avoids modeling errors
due to biased or incomplete knowledge. The number of open parameters is limited,
making the model relatively computationally efficient. In other words, input-output
model provides is well balanced between the common modeling dilemma: accuracy and
efficiency.

1.2.4. Discussion
Different approaches to model neuronal transformations have different advantages
and disadvantages. It is important to define the purposes of the research, in order to select
an appropriate approach.
Compartmental modeling is the best modeling methodology in interpreting
underlying mechanisms. Nevertheless, a real neuron is an extremely complex entity and it
is not impossible to simulate it in every detail. Thus, certain type of simplifications are
always necessary according to the modeling purposes. During the process of
simplification, many assumptions must be made. Therefore, modeling errors due to
biased and incomplete knowledge are almost inevitable. The open parameters of
compartmental modeling are often redundant, making the mathematical optimization
difficult. Taking HH model for instance, there would be multiple ways to achieve higher
initial rising rate of PSP waveform, such as decreasing the capacitance; decreasing the
13

leaking conductance; accelerating sodium conductance dynamics; or having slower
potassium conductance dynamics.
Integrate and fire modeling methodology presents neuron in a highly simplified
format. This makes the method computational efficient and well serves the purpose of
building large-scale simulation. The over-simplification, however, makes the
methodology impossible to provide biological implications and quantitative simulations.
The applied input-output methodology implemented using Volterra series require
minimum assumptions and captures the input-output transformations based on recorded
intracellular activity (data-driven). This avoids the modeling errors due to biased or
partial knowledge. All model parameters are optimized according to rigorously defined
error terms, and no arbitrary manipulation is involved. The estimation and prediction
process can both be done with a regular personal computer (AMD Phenom 9750).
The single neuron model developed in this thesis is a hybrid, combining both
mechanistic and input-output modeling methodologies. The signal generation principles
common to all spike-input, spike-output neurons define the model structure. The model
structure is sufficiently general and can be applied to all spiking neurons.
14

1.3. Hippocampus CA1 pyramidal neurons and its short- and long-term plasticity
1.3.1. Functions of the hippocampus
Investigation of the functions of the hippocampus has lasted for over a century
and is still going on. Until the 1930s the hippocampal formation was considered by
neuroanatomists to be part of the olfactory system [10-12]. In late 1930s another
influential hypothesis was proposed, W. Papez (1937) suggested that the hippocampus
was part of a circuit that provides emotion. The significance of this hypothesis is that it
suggested that the hippocampus might have functions other than olfaction, even though
the hypothesis was based on no empirical evidence. Moreover, Richard Jung and Alois
Kornmuller (1938) found evidence that the hippocampus was involved in general
attention control.
Since late 20th century, memory formation is the most prominent function of the
hippocampus supported by direct human evidence. The idea that the hippocampus is
intimately associated with memory is due to observations of brain-damaged patients
observed by William Scoville and Brenda Milner in 1957. Scovill removed the mesial
aspects of the temporal lobes from several patients in an attempt to relieve a variety of
neurological and psychiatric conditions. The most famous of these, H.M., was a severe
epileptic patient whose seizures were resistant to antiepileptic drug treatment. Following
the surgery to remove the hippocampus, his seizures were reduced, but he was left with a
profound permanent global amnesia. H.M. could remember items for brief periods (10s
of seconds to few minutes), provided he was allowed to rehearse and is not distracted.
15

Upon distraction, however, H.M. rapidly forgot, meaning that he cannot form new long-
term memory. He reports his conscious existence as that of “constantly waking from a
dream and everything looks unfamiliar”. Much research confirm the point that the
hippocampus plays a central role in learning and memory [13]. Today, more detail
investigations are still going on to examine different functions of the hippocampus.

1.3.2. General introduction to CA1 pyramidal neurons
As shown in Fig. 1.4 the hippocampus formation, the entorhinal cortex is
considered the first step in the intrinsic hippocampal circuit.

Fig. 1.4. (a) Hippocampus neuroanatomy. (b) CA1 pyramidal neuron morphology.
16

The projections from the entorhinal cortex (EC) to the dentate gyrus (DG) form part of
the major hippocampal input pathway called the perforant path (PP). The connection
between EC and DG is unidirectional, meaning that the EC projects to the DG but the DG
does not project back to the EC. Likewise, the principal cell of the DG, the granule cell,
gives rise to axons called mossy fibers that connect with pyramidal cell of the CA3 field
of the hippocampus. The projection from the DG to the CA3 is largely unidirectional [14].
The pyramidal cell of CA3, the major source of the CA1 input, project to CA1 pyramidal
cell through axon fibers called Schaffer collaterals (SC) unidirectionally. The CA1
pyramidal cell then projects to subiculum unidirectionally.
For over a decade, CA1 pyramidal cells are the most studied sub-region of the rat
hippocampus [15]. Two elaborately branching dendritic trees emerge from the pyramid
shaped soma of CA1 neurons in the rat hippocampus (Fig. 1.4b). The basal dendrites
occupy the stratum oriens, and the apical dendrites occupy the stratum radiatum
(proximal apical) and the stratum lacunosum-moleculare (distal apical). The distance
from the stratum pyramidale to the hippocampal fissure (hf) is about 600 μm, and the
distance from the stratum pyramidale to the alveus is about 300 μm, yielding a distance
of about 1mm from tip to tip. CA1 pyramidal neurons are covered with about 30,000
dendritic spines. Electron microscopic, immunocytochemical, and physiological analyses
have all converged on the conclusion that most dendritic spines receive excitatory
synaptic inputs, indicating that spine density can be used as a reasonable measure of
excitatory synapse density [16-18]. The density of dendritic spines and synapses on CA1
17

pyramidal neurons is highest in the stratum radiatum and stratum oriens but lower in the
stratum lacunosum-moleculare [18-21].
The principal excitatory inputs to CA1 pyramidal cells arrive from the CA3
through SC and the EC through PP fibers. The inputs from CA3 pyramidal neurons
through SC form synapses on the apical dendrites in the stratum radiatum and on the
basal dendrites in the stratum oriens [22, 23]. The PP input from the EC to CA1
selectively innervates the distal apical dendrites in the stratum lacunosum-moleculare
[24]. There are identified differences between PP and SC input [25]. The PP synapses are
located farther from the soma, without a mechanism for compensating for this dendritic
disadvantage. These PP synapses innervations from EC would be expected to have a
weaker influence on action potential initiation in the axon than the SC synapses,
suggesting that SC, the input stimulation site of the model developed in this thesis, is the
major input for CA1. However, Yeckel and Berger (1990) showed the axons arise from
the EC to the CA1 are sufficient to drive the pyramidal cells in CA1 [25].

1.3.3. Short-term plasticity
Short-term plasticity (STP) is a use-dependent alteration in synaptic strength over
a time scale of milliseconds to seconds. STP produces two transient effects: facilitation
(the presence of previous stimulations enhance the current postsynaptic response) and
depression (the presence of previous stimulations suppress the current postsynaptic
response). In 1980, It was first reported when pairs of stimuli are delivered to the
perforant path, the amplitude of the second response recorded in the molecular layer of
18

the dentate gyrus in vivo is typically facilitated at inter-stimulus intervals of 200 to 300
ms and depressed at longer intervals of up to a few seconds [26, 27]. Similar but not
exactly the same effects are reported in Schaffer collateral synapse in CA1 region by
Andersen (1960).
Many underlying mechanisms of STP have been inferred from numerous studies
involving various experimental techniques and preparations [28-30]. The most widely
accepted theory suggested by analysis of elementary (quantal) synaptic events indicate
that STP is mainly due to presynaptic mechanisms [29, 31-33]. According to the
hypothesis, facilitation is caused by the accumulation of residual calcium in the
presynaptic bouton, while depression is due to the depletion of the neurotransmitter
vesicle pools. In the case of facilitation, each individual presynaptic spikes activates the
calcium channel, hence transiently increases the axonal calcium concentrations over a
time span of hundreds of milliseconds. Increased calcium concentration enhances the
probability of neurotransmitter vesicle release. In the case of depression, the presynaptic
ready-to-release pool of vesicles is reduced due to previous activities resulting in sort-
term depression. Although the presynaptic theory of STP is widely accepted, some
literature reports that postsynaptic nonlinear properties may also contribute to STP [34].
In this thesis, STP dynamics are captured and represented within the memory
window (1000mS) of Volterra kernels with the latency τ; see section 2.4.3. for detail
explanations.
19

1.3.4. Long-term plasticity
Long-term plasticity means long-term change in synaptic connection strength
over the time scale from hours to permanent. Long-term potentiation (LTP) and long-
term depression (LTD) are two common forms of long-term plasticity. In 1973, Bliss and
Lomo’s discovery of long-term potentiation in rabbit hippocampal DG area changes the
understanding of learning and memory forever. Recently, direct evidences showed that
LTP intimately involves in learning and memories [35-37]. Whitlock et al. (2006)
recorded field potentials from multiple sites in hippocampal area CA1 before and after
single-trial inhibitory avoidance learning [36]. Field potentials increased on a subset of
the electrodes, and these could be specifically related to the learning event. Pastalkova et
al. (2006) reversed hippocampal LTP in freely moving animals using a cell-permeable
inhibitor of a protein kinase [37]. Reversal was accompanied by a complete disruption of
previously acquired long-term memory in a place avoidance task, even when the kinase
inhibitor was infused only during the consolidation interval. This result suggests that LTP
was necessary for storing spatial information.
Generally speaking, LTP consists of two phases: early phase (minutes to hours;
kinase dependent) and late phase (hours to weeks; gene expression and protein synthesis
dependent). The induction mechanisms of long-term plasticity of SC-CA1 are different
from mossy fibers-CA3. This study focuses on the Schaffer collateral synapse (SC to
CA1). The activation of synaptic ion channel N-methyl D-aspartate (NMDA) type of
glutamate receptor is necessary for the initiation of LTP in Schaffer collateral [38-41].
20

LTP in the Schaffer collateral typically requires activation of several afferent axons
together, a feature called cooperativity. This feature derives from the fact that the
NMDA receptor channel becomes functional and conducts Ca
2+
only when two
conditions are met: glutamate must bind to the postsynaptic NMDA receptor and the
membrane potential of the postsynaptic cell must be sufficiently depolarized by the
cooperative firing of several afferent axons to expel Mg
2+
from the mouth of the channel.
Only when Mg
2+
is expelled can Ca
2+
influx into postsynaptic cell and hence activate
Ca
2+
dependent proteins and kinase which in turns initiate the early phase LTP. Beside
cooperativity, LTP in the Schaffer collateral pathway requires concomitant activity in
both the presynaptic and postsynaptic cells to adequately depolarize the post-synaptic cell,
a feature called associativity. As we have seen, to initiate the Ca
2+
influx into
postsynaptic cell, a strong presynaptic input sufficient to fire the postsynaptic cell is
required.
Even though the complete mechanisms of the maintenance of early phase LTP
remained unclear, at least four features are identified: 1. increase in sensitivity of alpha-
amino-3-hydroxy-5-methyl-4-isoxazolepropionic acid (AMPA) receptors [42-44]; 2.
increase in numbers of AMPA receptors [45, 46]; 3. increase in strength of presynaptic
vesicle release [47, 48]; 4. increase in spine conductance [49]. High level of internal Ca
2+

activates calcium/calmodulin-dependent protein kinase II (CaMKII), as important kinase
involving in LTP. Activated CaMKII phosphorylates AMPA receptor and increase its
sensitivity to glutamate. CaMKII might also be a modulator in new AMPA receptor
insertions. The autophosphorylation of CaMKII might also provide a mechanism to
21

maintain LTP beyond initial high Ca
2+
initiation period as being self sustainable [50].
Quantal analysis of Schaffer collateral synaptic transmission suggests that the strength of
presynaptic vesicle releases is substantially increased either due to increase in release
probabilities or vesicle size or both. Moreover, certain Ca
2+
dependent proteins,
specifically CaMKII and calpain, seems to play important roles in modifying
cytoskeleton structure of synapses through widen the relatively narrow spine neck to
increase the electro conductivity [51].
Unlike early phase LTP, which is independent of protein synthesis, late phase
LTP requires gene transcription and protein synthesis in the postsynaptic cell. The
mechanisms involved in late phase LTP seems similar in different synapses, not like early
phase LTP, which is synapse specific. Two phases could be further categorized for late
phase LTP: the first depends upon protein synthesis, while the second depends upon both
gene transcription and protein synthesis. Many of the mechanisms of late phase LTP still
remain unclear, however, some features are identified. First late phase LTP consists of
local protein synthesis (gene expression independent) resulting in local synapse/spine
modification; second late phase LTP, on the other hand, requires gene transcription,
specifically cAMP-PKA-MAPK-CREB signaling pathway, resulting in forming new
synaptic connections [41, 52-54].
The knowledge of how long-term plasticity affects kernels provide the basis for
developing adaptive large-scale simulation of neuronal networks and adaptive cortical
prosthesis. Further, the kernel after the induction of long-term plasticity might keep
changing. Tracking the kernel change may help to identify the temporal order of different
22

LTP mechanisms being introduced. Also, it has been suggested that different LTP
induction protocols may activate distinct signaling cascades that generate LTP with
different expression mechanisms [55-57]. If the hypothesis is true, by applying different
LTP induction procedure, we should see additive effect of LTP expression and the
kernels analysis should reveal the effects.
23

1.4. Whole-cell patch clamp
1.4.1. Introduction
A profound advancement in electrophysiology came with the development of the
gigaseal and patch-clamp methods. Erwin Neher and Bert Sakmann wanted to record
from a tiny area (a patch) of surface membrane by pressing a fire-polished pipette against
a living neuron. In 1976 they reported the first single channel current record with an
acetylcholine-activated channel. But the real breakthrough was reported in 1981, when
they showed that clean glass pipettes can fuse with clean cell membranes to from a seal
of unexpectedly high resistance and mechanical stability [58]. They called the seal a
“gigaseal” since it could have an electrical resistance as high as tens of gigahoms. Erwin
Neher and Bert Sakmann shared the Nobel Prize in Physiology or Medicine in 1991 for
this technique.
The gigaseal permitted four electrical recording configurations (see Fig. 1.5).

Fig. 1.5. Four gigaseal recording methods (Copied from Ion channels of excitable membranes, Hille, p.88).
24

As soon as the pipette is sealed to the cell membrane, one can record single channels
under voltage clamp in the on-cell mode. The seal is so stable that the patch can be pulled
off the cell and still sealed to the pipette and dipped into a variety of test solutions- the
inside-out configuration. Alternatively, an on-cell patch may be deliberately ruptured by
suction with the pipette keep sealing to the cell, In that case one is recording in the
whole-cell configuration. Finally, pulling the pipette away from the cell in whole-cell
mode results in an outside-out patch. Almost at once these recording methods became the
primary techniques of membrane biophysics, and because the necessary equipment was
soon commercially available, voltage clamping also became practical for neurobiologists.
The major advantage of the whole-cell patch clamp is that it offers superior ability
to control either the membrane voltage (voltage clamp) or membrane current (current
clamp) [7]. It offers scientists tremendous opportunities to investigate specific voltage
dependent channels, ligand binding channels, and neurotransmitter vesicles through
combining whole-cell patch clamp with proper pharmaceutical manipulation and
experimental design.

1.4.2. Circuits
The voltage clamp and current clamp are done with a current to voltage converter.
Fig. 1.6 shows a current to voltage converter demonstrating the electrical principle behind
whole-cell recording [59].
25

Fig. 1.6. Electronic scheme of current to voltage converter for whole-cell recording.

In this scheme, V
m
represents membrane potential (inside), V
ref
represents the
extracellular potential (outside), V
i
represents Op Amp output voltage, and I
p
represents
the pipette current. The relationship among them is given
mi p
VV IR  (1.22)
In whole-cell patch clamp, we could either fix V
m
and let neuron membrane current drive
the pipette current I
p
(voltage clamp), or fix I
p
and let V
m
fluctuate with the neuron
membrane voltage.
26

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31

CHAPTER 2: HIGH-ORDER DYNAMICAL MODELING OF SINGLE
NEURONS WITH CONSTANT THRESHOLD

2.1. Short introduction
My goal, in this chapter, is to develop a single neuron model which 1) can capture
nonlinear dynamics of single neuron sub-threshold (postsynaptic potentials, PSPs) and
supra-threshold (action potentials) activities; 2) has a general structure and thus can be
applied to any spike-input, spike-output neuron; 3) is computationally efficient.
The transformation of spike trains between neurons is a complex process that
involves calcium-influx in presynaptic terminal, presynaptic vesicle release of
neurotransmitter, postsynaptic transduction, synaptic integration, somatic integration and
action potential formation, action potential back-propagation into the dendritic arbors,
and retrograde signaling towards the presynaptic terminal. All these mechanisms
interactively contribute to the nonlinear and dynamical characteristics of neuron-to-
neuron spike train transformation. To characterize this transformation, the concerns are
two-fold. First, how to design an experimental paradigm to collect information-rich
input-output data sets that contain interactions among the mechanisms mentioned as
many as possible? Second, how to construct a model that can characterize neuron
nonlinear input-output transformation based on the experimental data with minimum
assumptions to avoid the bias of partial knowledge?
To meet these requirements, broadband stimulation trains of single all-or-none
pulses were delivered to the synaptic region of CA1, the stratum radiatum containing
32

Shaffer collaterals (SCs) (see Fig. 2.1A). The all-or-none stimulation pulses mimic the
action potentials, the most common form of input and output signal in biological neural
systems. The inter-pulse-intervals of the stimulation trains follow a broadband
distribution. These broadband stimuli can elicit broad range of physiological responses
and nonlinearities that are resulted from the interactions of the sub-cellular neuron
processes mentioned above [1-4]. This paradigm mimics the natural condition in
biological nervous system and elicits most of the neuronal processes mentioned above as
opposed to other simpler paradigms, e.g., somatic current injection stimulation paradigm,
which involves a smaller subset of all possible mechanisms.

33

Fig. 2.1. Experimental Paradigm. A: Schematic diagram of a hippocampal slice. The bipolar stimulation
electrode was placed on the CA1 stratum radiatum. Poisson random interval stimulation trains were
delivered through a stimulation electrode. A recording glass electrode was patched on the soma of a CA1
pyramidal neuron to record the elicited PSPs and action potentials. B: Picture showing a hippocampal slice
with stimulating and recording electrodes. C: Picture showing a recording electrode patching on the soma
of a CA1 pyramidal neuron.

34

In this chapter, whole-cell patch-clamp was performed to record corresponding
intracellular PSPs and action potentials, because it provides a high-quality continuous
tracing of neuron intracellular signal in both sub-threshold and supra-threshold response
regimes. The SC-CA1 pyramidal cell system was chosen for the study because it is one of
the most studied brain region.
The model structure was constructed based on three principal neuronal processes
that are common in all spike-input, spike-output neurons (Fig. 2.2): 1) transformation
from presynaptic spike to sub-threshold PSP, 2) action potential generation, and 3) spike-
dependent modification of sub-threshold membrane potential.
35

Fig. 2.2. The model comprises three major components: feedforward Volterra kernels (k), a threshold (  ,
and a feedback Volterra kernel (H). The feedforward kernels are up to third order, describing the
transformation from presynaptic stimulation (x) to sub-threshold PSP (u). The threshold ( ) is a constant
membrane potential level, above which output action potentials (yh) are generated. The pre-threshold
membrane potential (non-spiking) (w) is the summation of sub-threshold PSPs (u) and spike-triggered
after-potentials (a). The operation  is defined as superimposition. The model output (y) is the
superimposition of pre-threshold membrane potentials (w) and templates of action potentials.

The signal flow of the model starts from the entering of presynaptic spikes to the
feedforward blocks (Fig. 2.2), which characterize the nonlinear transformation from
presynaptic spikes to sub-threshold PSPs; sub-threshold PSPs then are passed to a
constant threshold, above which action potentials are formed; output action potentials
then trigger a feedback block, which describes the dynamics of spike-triggered after-
potential that in turn modifies the membrane potential following each action potential. In
36

our approach, both feedforward and feedback blocks were implemented with Volterra
models [5, 6]. In a Volterra model, the output signal is expressed in terms of the input
signal by means of a Volterra power series. The input-output nonlinear dynamics of the
system is described by a series of progressively higher order Volterra kernels, which can
be directly estimated from the experimental input-output data. This data-driven property
avoids the modeling errors caused by unknown mechanisms and/or partial knowledge of
the system.
37

2.2. Experimental procedures
Rat hippocampal slices were prepared acutely before each experiment. Two-
week-old male Sprague-Dawley rats were anesthetized with inhalant anesthetic,
Halothane (Halocarbon Laboratory, NJ), before standard slicing procedures.
Hippocampal slices (400μm) were prepared using a vibrotome (Leica VT 1000s,
Germany) under iced sucrose solution. A surgical disruption of the connection between
CA3 and CA1 was performed on each slice before it was transported to oxygenated bath
solution for maintenance at 25˚C. The sucrose solution contained (in mM): Sucrose 206,
KCl 2.8, NaH
2
PO
4
1.25, NaHCO
3
26, Glucose 10, MgSO
4
2, Ascorbic Acid 2; at pH 7.5
and 290 mOsmol. The bath solution contained (in mM): NaCl 128, KCl 2.5, NaH
2
PO
4

1.25, NaHCO
3
26, Glucose 10, MgSO
4
1, Ascorbic Acid 2, and CaCl
2
2; at pH 7.4 and
295 mOsmol.
The experiments were performed on the hippocampal SC-CA1 system. CA1
neurons in the rat hippocampus have two elaborately branching dendritic trees (basal and
apical) emerge from the pyramid-shaped soma (Fig. 2.1A). The basal dendrites occupy
the stratum oriens, and the apical dendrites occupy the stratum radiatum (proximal) and
the stratum lacunosum-moleculare (distal). The distance from the stratum pyramidale to
the hippocampal fissure (hf) is about 600 μm, and the distance from the stratum
pyramidale to the alveus is about 300 μm, yielding a 1mm distance from end to end. CA1
pyramidal neurons are covered with about 30,000 dendritic spines. The principal
excitatory inputs to CA1 pyramidal cells arrive from CA3 through SC. The inputs from
38

CA3 pyramidal neurons through SC form synapses on the apical dendrites in the stratum
radiatum and on the basal dendrites in the stratum oriens [7, 8].
During whole-cell patch-clamp recording, hippocampal slices were perfused with
oxygenated bath solution at 25˚C. A bipolar stimulation electrode was placed in the CA1
stratum radiatum according to visual cues. Bipolar stimulation electrodes were made in
laboratory with formvar-insulated Nichrome wire (A-M Systems Inc., WA). A recording
mirco-pipette electrode with 4 M Ω tip resistance patched on the somatic membrane of
CA1 pyramidal neuron to record the intracellular PSPs and action potentials. The
recording micro-pipette electrodes were produced by heating and pulling thin-wall single
barrel borosilicate glass tubing (World Precision Instrument, FL) using a pipette puller
(Sutter Instrument P-80/PC). The internal solution of the recording electrode contained
(in mM): Potassium-Gluconate 110, HEPES 10, EGTA 1, KCl 20, NaCl 4, Mg-ATP 2,
and Na
3
-GTP 0.25; at pH 7.3 and 290 mOsmol.
A programmable stimulator (Multi Channel System, Germany) was used to
deliver Poisson random interval trains (RITs) with a 2 Hz mean frequency with inter-
spike-intervals ranging from 10 to 4500 ms [9-11]. Whole-cell patch-clamp recording in
the mode of current clamp were performed with the HEKA EPC9/2 amplifier with 10
kHz sampling rate. The intensities of the stimulation spike trains were adjusted so that
approximately 50% of the stimulations induced action potentials. This study included 98
trials of 200 second recordings (400 stimulations in each recording trial) in 15 cells from
13 different animals.
39

2.3. Model structure and its mathematical expressions
Sub-threshold dynamic, spike generation, and spike-triggered after-potential
dynamic are separated in the model structure in a physiologically plausible manner and
are captured by an up to third-order feedforward kernel, a constant threshold, and a first-
order feedback kernel, respectively (Fig. 2.2). The feedforward kernels describe the
nonlinear dynamical effects of synaptic transmission, dendritic integration, and somatic
integration and transform presynaptic spikes to PSPs [12-14]. If a corresponding PSP
response is higher than the threshold, a template waveform of an action potential is
superimposed to the sub-threshold membrane potential. The formed action potential then
triggers the feedback kernel and generates a spike-triggered after-potential that modifies
the following membrane potential [15-21].
The model (Fig. 2.2) can be expressed with the following equations:
(, ) ( , ) wuKx aHyh  (2.1)
action potential ,
,
w
y
ww


 




(2.2)
In (1), w represents the pre-threshold (non-spiking) membrane potential, which is the
summation of the output of the feedforward block, u, and the output of the feedback
block, a. The feedforward block, K, transforms the presynaptic spike-trans, x, to sub-
threshold PSPs, u. The feedback block, H, describes the transformation from the output
spikes, yh, to the spike-triggered after-potentials, a. In (2), y represents the output of the
neuron model, a continuous trace of predicted sub-threshold and supra-threshold
40

membrane potentials. If w is higher than or equal to the threshold, , a template of an
action potential is superimposed to w, and the feedback kernel is triggered by the output
action potentials, yh; if w is lower than , y is equal to w.
The output of the feedforward block, u, is expressed with an up to third-order
Volterra model as,
1
12
12 3
011 1
0
21 2 1 2
00
31 2 3 1 2 3
00 0
() ( ) ( )
( , )( )( )
(, , ) ( ) ( ) ( )
k
kk
kk k
M
MM
MM M
ut k k x t
kxt xt
kxtxtxt


 

  
    


 
 




 
(2.3)
In (3), k
0
is the zero-order kernel, describing the output of the system output when the
input is absent, i.e., the resting membrane potential. The first-order feedforward kernel, k
1
,

describes the system’s first-order (but not single-pulse) response to x (see the definitions
of response functions in Table 1 for more explanations). The second-order feedforward
kernel, k
2
, describes the second-order (but not paired-pulse) response to x. The third-order
feedforward kernel, k
3
, describes the third-order (but not triple-pulse) response to x. M
k
is
the length of the memory window.
The spike-triggered after-potential, a, in (1), is expressed with a first-order
Volterra model as,
1
() ( ) ( )
h
M
at h yh t

 



(2.4)
where the feedback kernel, h, describes how an output action potential, yh, triggers an
after-potential. The output action potential train, yh, was defined as,
41

1 ,
0 , <
w
yh
w


 



(2.5)
The Laguerre expansion method provides a practical method to reduce the
number of open parameters in Voltera kernel [22]. In this chapter, both feedforward
kernel, K, and feedback kernel, H, are expanded using orthonormal Laguerre basis
functions b [6].
42

2.4. Model estimation procedures
The Laguerre expansion method effectively reduces the number of open
parameters in the Volterra model by expanding the Volterra kernels with Laguerre basis
functions [6, 22, 23]. Both feedforward and feedback kernels are estimated using the
Laguerre expansion method as,
01
1
21 2
12
12
3123
12 3
1
12
11
12 3
11 1
() ( ) ()
(, ) () ()
( , , ) () () ()
L
k
kk j
j
j L
kk
kjj
jj
jj L
kk k
kjjj
jj j
ut c c j v t
cj j v tv t
cj j jv tv tv t


 





 
(2.6)
where,
0
() ( ) ( )
k
M
k
jj
vt b xt

 



(2.7)
In (6), L denotes the number of Laguerre basis functions;
0
k
c ,
1
k
c ,
2
k
c ,and
3
k
c are the
Laguerre expansion coefficients of the feedforward kernels, k
0
, k
1
, k
2
, and k
3
, respectively;
()
k
j
vt are the convolutions of the Laguerre basis functions, b
j
, and the input stimulation
train, x.
Similarly,
1
() ( ) ()
L
h
hj
j
at c j v t



(2.8)
where,
43

1
() ( ) ( )
h
M
h
jj
vt b yht

 



(2.9)
In (8), L denotes the number of Laguerre basis functions; c
h
denotes the Laguerre
expansion coefficients of the feedback kernel, h. In (9), ()
h
j
vt are the convolutions of the
Laguerre basis functions, b
j
, and the output action potentials, yh. Laguerre basis functions
b
j
is expressed as,
()/2 1/2
0
()/2 1/2
0
(1) (1 )
(1) (1 ) , (0 )
()
(1) (1 )
(1) (1 ) , ( )
j
kk k
k
j
jj
j
kjk k
k
j
j
kk
b
j
j M
kk






 



 










   










   






(2.10)
where  is the time epoch value; j is the number-order of basis functions;  is the
Laguerre parameter (0< <1) determining the rate of exponential asymptotic decline of
the Laguerre basis functions.

2.4.1. Parameter optimizations and estimations
In this model, , L, c
k
, c
h
, and  are the open parameters that need to be estimated
according to either the error measures. The optimization processes are discussed in the
following.
44

2.4.1.a. Optimization of Laguerre parameters
Laguerre parameter, , determines the rate of the exponential asymptotic decline
of the Laguerre basis functions. For each in-sample-training trial, I tried 0.5-0.99 to find
the minimum NMSE and applied the Quasi-Newton method [24] to refine the optimal 
values for feedforward kernels ( 
k
) and feedback kernel ( 
h
). The optimal 
k
and 
h

produce a global minimum of NMSE in fitting the recorded data (Fig. 2.3A).
45

Fig. 2.3. A1: A contour plot of NMSE with respects to the Laguerre parameters of the feedforward kernels
and the feedback kernel 
k
and 
h
. The two dash lines intersecting at the global minimum of NMSE are
plotted in A2 and A3. The arrows in A2 and A3 indicate the global minimum of NMSE. In this example,
the NMSE converged to the global minimum value 12.648% at 
k
= 0.972 and 
h
= 0.910. B: choosing the
optimal number of Laguerre basis functions (L). In the in-sample training trial (dashed line), the NMSE
decreased monotonically with increasing numbers of basis functions. In the out-of-sample prediction trials
(solid line), the minimum NMSE was found at L = 3.
46

2.4.1.b. Optimization of the total number of Laguerre basis functions L
The relation between L and NMSE is evaluated in Fig. 2.3B. In the case of in-
sample trainings, the NMSE decreases monotonically as the number of basis functions
increases. However, in the out-of-sample predictions, the NMSE reaches the minimum at
L=3, and rises slightly afterwards. This phenomenon is known as the multi-variable over-
fitting effect. In other words, when the number of basis functions is higher than three, it
produces over-fitted results that are specific to the training dataset and loses the model
generality [25]. The analysis indicates that three is the optimal number for basis functions.

2.4.1.c. Estimation of Laguerre expansion coefficients c
k
and c
h

The optimal Laguerre expansion coefficients c
k
, and c
h
are estimated using the
least-squares method as the product of the pseudoinverse matrix of V and the recorded
data, ỹ (after removing artifacts and action potentials).



0
T1T
(1 )
1
(1 ) 1
(1)
(1)
(2)
()
()
(1)
()
()
k
k
k
pq T
h
T
h
pq
c
c
y
y
VV V
cp
c
yT
cq

 

 



 

 

 
 

  

 

 
 






(2.11)
In (11), T is the data length of the recorded data; 1+p is the total number of feedforward
convolutions
1
k
j
T
V



and their products; q is the total number of feedback convolutions
47

1
h
j
T
V



; V is the concatenated matrix of all
k
j
V ,
h
j
V , and their applicable products,
expressed as,
112 21 2 1 31 2 1 3 2
11 , 1 1 1, 1 , 1
[1 , , , , ]
kkk kkk h
jjj j jL j j L j j j jL jj L j j j j
V V VV VV V V
          


(2.12)
In (11),
T
V refers to the transpose matrix of V. See Fig. 4A for scatter plot comparison of
recorded PSP and predicted PSP.

2.4.1.d. Optimization of constant threshold 
The thresholds are estimated by scanning through the range 0~20mV (step size
0.01mV) to find a threshold value that gives the least SPER. Receiver operating
characteristic (ROC) curves (Fig. 2.4B) are plotted with respect to the false positive rate
(x axis) and true positive rate (y axis). The optimal threshold has a closest L1-distance to
the upper left corner (0, 1) in the ROC curve.
48

Fig. 2.4. Evaluations of model prediction performance with normalized mean square error (NMSE) and
spike prediction error rate (SPER). A: scatter plot of recorded data and predicted data. In this example,
NMSEs of K
1
, K
2
, and K
3
models were 18.3%, 15.4%, and 14.1% respectively (only K
3
result is plotted). B:
estimating the optimal threshold with a receiver operating characteristic (ROC) curve. In this example,
SPERS of K
1
, K
2
, and K
3
models were 28.2%, 23.6%, and 20.1% respectively.

2.4.2. Model interpretations
2.4.2.a. Reconstructions of Volterra kernels
The feedforward and feedback Volterra kernels are reconstructed using the
optimal Laguerre expansion coefficients c
k
and c
h
and the Laguerre basis functions b
j
as
follows [26]:
49

11
1
11
1
() ( ) ()
L
kj
j
kcjb  



(2.13)

1
2
12 2 1
12
12
21 2 1 2 1 2
11
(, )
(, ) ( ) ( ) ( ) ( )
2
j L
k
jj j j
jj
cj j
kbbbb 



(2.14)
12 3
13 2
12
21 3
3
12 3 23 1
31 2
32 1
123
12 3
12 3
12 3
31 2 3
11 1 12 3
12 3
12 3
() ( ) ( )
() ( ) ( )
() ( ) ( )
(, , )
(, , )
() ( ) ( ) 6
() ( ) ( )
() ( ) ( )
jj j
jj j
jj L
jj j
k
jj j jj j
jj j
jj j
bb b
bb b
bb b
cj j j
k
bb b
bb b
bb b
 
 
 
 
 
 
 
 
















 
(2.15)
1
() ( ) ()
L
hj
j
hcjb  



(2.16)

2.4.2.b. Calculation of response functions
To examine the effect of a given number of input pulses, I utilized the notion of
response functions [5, 25]. As shown in Table 2.1, response functions (r) can be easily
calculated from Volterra kernels (k). The first-order response function r
1
describes the
single-pulse response elicited by a single input pulse; the second-order response function
r
2
describes the paired-pulse effect caused by pairs of input pulses; the third-order
response function r
3
describes the triple-pulse effect caused by triplets of input pulses.
The feedback Volterra kernel is only of first order, thus the feedback response function is
equal to the feedback Volterra kernel (h), as in (16). Examples of response functions are
plotted in Fig. 6, 7, and 8.

50

K
1
K
2
K
3

r
1 11
() () rk   
11
2
() ()
(, )
rk
k
 
 



11
2
3
() ()
(, )
(, , )
rk
k
k
 
 
 




r
2
N/A
21 2 2 1 2
(, ) 2 (, ) rk  
21 2 2 1 2
31 1 2
32 2 1
(, ) 2 (, )
3( , , )
3( , , )
rk
k
k

 
 




r
3
N/A N/A
31 2 3 31 2 3
(, , ) 6 ( , , ) rk      

Table 2.1. Conversions between the feedforward Volterra kernels (k
1
, k
2
, and k
3
) and the feedforward
response functions (r
1
, r
2
, and r
3
) of the K
1
, K
2
, and K
3
models. The single-pulse (r
1
), paired-pulse (r
2
), and
triple-pulse (r
3
) response functions combine the contributions of identical stimulations described among
different orders of Volterra kernels (k
1
, k
2
, and k
3
) to facilitate physiological interpretations. Examples of
the response functions are plotted in Fig. 6, 7, and 8.

2.4.2.c. K
1
model
The response functions (r
1
and h) of a representative K
1
model are plotted in Fig.
2.5. In Fig. 2.5A, r
1
rises abruptly in the beginning from -0.6 mV, reaches the peak
amplitude 10.3 mV at = 24 ms, and decays to lower than 1 mV (less than 10% of the
peak) at 112 ms. In Fig. 2.5B, h has a peak amplitude of 6.3 mV at the beginning and
decays to lower than 0.6 mV (less than 10% of the peak) at 121 ms. The K
1
model's
average prediction performance over all datasets in NMSE is 17.9% and SPER is 22.4%.
51

Fig. 2.5. The response functions of a K
1
model. A: the single-pulse feedforward response function (r
1
) is
the single-pulse response to an input event. B: the feedback response function (h) describes the spike-
triggered after-potential. The insets illustrate the relations between input pulses, output spike, and the time
epoch .

2.4.2.d. K
2
model
The response functions (r
1
, r
2
, and h) of a representative K
2
model are plotted in
Fig. 2.6. In Fig. 2.6A, r
1
rises to the peak 10.3 mV at 20 ms and decays to 1 mV at 118
ms. In Fig. 2.6B, on the diagonal, r
2
starts from -6.17 mV, rises to a minimum 0 mV at
(60 ms, 60 ms), reaches the peak 1.79 mV at (90 ms, 90 ms), and decays to 0.14 mV at
(190 ms, 190 ms). In general description, r
2
response function is depressive for short-
interval (less than 60 ms) input pairs, facilitative for longer-interval (more than 60 ms)
input pairs, and relatively ineffective to input pairs that has an interval longer than 190
52

ms. In Fig. 2.6C, h has a peak amplitude 6.3 mV at 6 ms and decays to 0.6 mV at 33 ms.
The K
2
model's average prediction performance over all datasets in NMSE is 15.1% and
SPER is 20.2%.

Fig. 2.6. The response functions of a K
2
model. A: the single-pulse feedforward response function (r
1
). B:
the paired-pulse feedforward response function (r
2
). C: the feedback response function (h). The insets
illustrate the relations between input pulses, output spike, and the time epoch .

53

2.4.2.e. K
3
model
The response functions (r
1
, r
2
, r
3
, and h) of a representative K
3
model are plotted
in Fig. 2.7. In Fig. 2.7A, r
1
rises to the peak 10.39 mV at 22 ms and decays to 0.96 mV at
113 ms. In Fig. 2.7B, on the diagonal, r
2
starts negatively at -1.28 mV, reaches the
minimum -5.5 mV at (30 ms, 30 ms), rise to 0.3 mV at (70 ms, 70 ms), reaches the peak
2.3 mV at (110 ms, 110 ms), and decays to 0.19 mV at (220 ms, 220 ms). In Fig. 2.7C, r
3

starts at peak 12.6 mV at the beginning and decays to 0.95 mV in 50 ms. In general
description, r
3
is facilitative if the previous two pulses have intervals less than 60 ms to
the current pulse. The facilitative effect of r
3
become ineffective when the previous
pulses have intervals longer than 60 ms. In Fig. 2.7D, h has a peak amplitude 5.5 mV at
the beginning and decays to 0.54 mV at 50 ms. The K
3
model's average prediction
performance over all datasets in NMSE is 14.4% and SPER is 18.8%.
54

Fig. 2.7. The response functions of a K
3
model. A: single-pulse feedforward response function (r
1
). B:
paired-pulse feedforward response function (r
2
). C: the triple-pulse feedforward response function (r
3
). The
insets illustrate the relations between input pulses, output spike, and the time epoch .

55

2.4.3. Evaluations and quantifications of model prediction performance
The model parameters/coefficients are estimated according to two error measures:
1) normalized mean square error (NMSE), which was used to evaluate the sub-threshold
PSPs waveform prediction; 2) spike prediction error rate (SPER), which was used to
evaluate spike prediction. NMSE was defined as,


2
1
2
1
(() ())
NMSE
()
T
t
T
t
yt yt
yt






(2.17)
where ỹ is the recorded data; y is the predicted data; T is the total number of data
points in ỹ and y. In both ỹ and y, action potentials are excluded without loss of
generality.
SPER was defined as the total number of false predictions divided by the total
number of stimulations, expressed as,
Numbers of False-positive + Numbers of False-negative
SPER=
Total Number of Stimulations
(2.18)
Both false-positive and false-negative cases are false predictions made by the
model. False-positive refers to the situation that the model predicts a response event (a
PSP response evoked by a stimulation) to form an action potential, but the corresponding
response event does not form an action potential in recordings. False-negative refers to
the situation that the model predicts a response event not to form an action potential, but,
the corresponding event in recordings forms an action potential.

56

2.4.4. Out-of-sample predictions
The out-of-sample prediction is performed in two steps. First, the sub-threshold
PSP, u, is predicted as follows:
0
1
(1 )
1
(1 ) 1
(1)
(2) (1)
1
() ()
k
kk k
p
Tp
k T
p
c
u
u c
VV
uT cp


 






 





 


 
(2.19)
Secondly, each PSP response is checked consecutively in the time order to see if
it surpasses the threshold, . If it does, an action potential template is superimposed at the
time point to the surpassing membrane potential, and a spike-triggered after-potential, a,
is added to the subsequent pre-threshold membrane potential, w, forming a recurrent loop
in the prediction process (Fig. 2.2). All predictions presented in this thesis are out-of-
sample predictions, i.e., using one dataset in training, and another independent dataset in
prediction. The prediction processes are computationally efficient and were performed
with a PC (AMD Phenom 9750).

2.4.4.a. Representative prediction clips
The single-input, single-output model described in this chapter can capture high-
order nonlinear dynamical characteristics of the Schaffer collateral-CA1 system.
Representative recorded and predicted data are shown in Fig. 2.8. To demonstrate how
progressively higher order kernels contributed to the prediction accuracy, I built three
models for each dataset: K
1
includes an first-order feedforward kernel (k
1
), K
2
includes
up to second-order feedforward kernels (k
1
and k
2
), and K
3
includes up to third-order
57

feedforward kernels (k
1
, k
2
, and k
3
); all models contain a threshold ( ) and a first-order
feedback kernel (h).

Fig. 2.8. A representative clip of recorded data (the upper panel) and the corresponding predicted data (the
lower panel) with an optimal threshold (dashed line).

2.4.4.b. PSP predictions
Sub-threshold PSP prediction was evaluated with NMSE. The distribution of out-of-
sample NMSEs to all datasets are shown in Fig. 2.9 (N=98). The average NMSEs to K
1
,
K
2
and K
3
models are 17.9%, 15.1% and 14.4% respectively (Fig. 2.9A). Moreover, the
NMSE improvement from K
1
to K
2
is 14.2% and from K
1
to K
3
is 18.7% (Fig. 2.9B).

58

Fig. 2.9. Distributions of NMSEs and NMSE improvements with increasing model orders. A: NMSE
histograms of K
1
, K
2
, and K
3
. B: NMSE improvement histograms.
59

2.4.4.c. Spike predictions
Spike trains are predicted using the estimated kernels and the optimal thresholds as
described in the Methods section. The distribution of SPERs of all datasets are shown in
Fig. 2.10. The average SPERs to K
1
, K
2
and K
3
are 22.4%, 20.2% and 18.8% respectively
(Fig. 2.10A). The SPER improvement from K
1
to K
2
is 11.2% and from K
1
to K
3
is
18.7% (Fig. 2.10B).

60

Fig. 2.10. Distributions of SPERs and SPER improvements with increasing model orders. A: SPER
histograms of K
1
, K
2
, and K
3
. B: SPER improvement histograms.
61

2.5. Discussion
The nonlinear dynamical single neuron model described in this chapter is
genuinely data driven. In other words, all model parameters are simultaneously
constrained using experimental data (intracellular recording) with rigorously defined
error measures, i.e., NMSE and SPER. No arbitrary manipulation of model parameters in
regard to error terms is involved. More importantly, all data constraining the model
parameters are derived from a single experimental set for which broadband input
conditions are imposed on the preparation. As argued elsewhere [5, 27], these represent
ideal conditions for modeling input-output nonlinear dynamics of a neurobiological
system. Finally, the model captures sub-threshold and supra-threshold activities, and their
interactions, in a single mathematical formalism.
The model described here is a general one. It is a hybrid, combining both
mechanistic (parametric) and input-output (nonparametric) components. Principles of
neuronal processes common to all spike-input, spike-output neurons, e.g., biological
signal generation and flow (Fig. 2), determine the model structure. On the other hand, the
specific properties that are variable from neuron to neuron are captured and quantified
with descriptive model parameters, which are directly constrained by intracellular
recording data. This hybrid representation of both parametric and nonparametric model
components partitions data variance with respect to mechanistic sources and thus imposes
physiological definitions to the model components and facilitates the biological
interpretations of the parameters. In addition, this hybrid structure representation
62

conducts the estimation power of each model component to specific dynamics of the
designated neuronal processes, producing more accurate estimations as opposed to
capturing the neuron input-output transformation as a whole with a single nonparametric
model component. This model is general enough to be applied to any spike-input, spike-
output neuron, and flexible enough to capture neuron-to-neuron differences.
In general, achieving both high accuracy and high efficiency in a single model
formalism is difficult. High model accuracy usually is achieved by including a large
number of open parameters, though unfortunately, this often reduces computational
efficiency. High computational efficiency usually is achieved by simplifying the
modeling methodology; however, model accuracy is inevitably compromised. Our results
show that the model described in this chapter achieved both high accuracy and high
efficiency. The model accuracy in predicting sub-threshold PSP measured by NMSE and
supra-threshold spiking activity measured by SPER were both lower than 20%. Moreover,
the modeling procedures can be performed easily using a PC (AMD Phenom 9750), and
the total number of open parameters is relatively small, e.g., 10 open parameters in the
case of the first-order model, 16 in the case of the second-order model, and 26 in the case
of the third-order model.
The model shown here is an extension of an earlier approach developed by our lab
[26]. In that study, Song et al. demonstrated the feasibility of using this model structure
to capture the nonlinear dynamics when the input-output signals are extracellularly
recorded unitary activities (spikes). Due to the different modeling goals and data-
collecting paradigms, there are three major differences between the neuron model used
63

by Song et al. and the neuron model in this chapter: 1) the number of model inputs, 2) the
noise term before threshold, and 3) the parameter estimation method. First, the model by
Song et al. utilized extracellularly recorded ensemble spike trains, and thus was a multi-
input, multi-output (MIMO) model. Since each CA1 neurons can potentially receive
inputs from multiple CA3 neurons, it is reasonable to include all observed inputs in the
model and then down-select them with statistical methods [25]. By contrast, in the
experimental preparation of this study, recorded CA1 neurons received artificially
delivered stimulation pulses from one stimulation source. In other words, the synapses of
the intracellularly recorded CA1 neurons were activated simultaneously. This resulted in
a single-input, single-output (SISO) formalism.
Second, the extracellular MIMO model by Song et al. and the intracellular SISO
model developed here dealt with noise differently. There are two major sources of
stochastic activity in a neuron: 1) unobserved inputs, and 2) intrinsic randomness of
underlying mechanisms [28-30]. Both sources exist in the experimental context of the
extracellular MIMO model. Thus, it is necessary for the MIMO model to include a noise
term to capture those stochastic activities. By contrast, in this chapter, the synaptic
connections between CA3 and CA1 neurons were surgically eliminated, so that no
spontaneous activity occurred. This experimental preparation drastically reduced the
stochastic level of the recorded CA1 neuron system, and resulted in a model structure
without an explicit noise term.
As to the last difference in parameter estimation method, extracellularly recorded
all-or-none action potentials do not provide direct information about sub-threshold
64

membrane potential. This lack of sub-threshold information along with the introduction
of a noise term led the MIMO model to the utilization of a maximum-likelihood method
for parameter estimation. On the other hand, intracellular whole-cell patch-clamp traces
continuously the membrane potential in both sub-threshold and supra-threshold regimes.
This information-rich data along with a low noise level enabled the use of least-squares
estimation to estimate the model parameters in this chapter.
65

2.6. References

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66

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[13] R. S. Zucker, and W. G. Regehr, “Short-term synaptic plasticity,” Annu Rev
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[14] P. Poirazi, T. Brannon, and B. W. Mel, “Arithmetic of subthreshold synaptic
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[15] R. Jolivet, T. J. Lewis, and W. Gerstner, “Generalized integrate-and-fire models
of neuronal activity approximate spike trains of a detailed model to a high degree
of accuracy,” J Neurophysiol, vol. 92, no. 2, pp. 959-76, Aug, 2004.

[16] B. E. Alger, and R. A. Nicoll, “Pharmacological evidence for two kinds of GABA
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[17] T. W. Berger, G. Chauvet, and R. J. Sclabassi, “A biologically based model of
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[18] W. Gerstner, and W. Kistler, Spiking neuron models: single neurons, populations,
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[19] J. Keat, P. Reinagel, R. C. Reid et al., “Predicting every spike: a model for the
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[20] L. Paninski, J. W. Pillow, and E. P. Simoncelli, “Maximum likelihood estimation
of a stochastic integrate-and-fire neural encoding model,” Neural Comput, vol. 16,
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[21] I. Song, and R. L. Huganir, “Regulation of AMPA receptors during synaptic
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67

[23] T. P. Zanos, S. H. Courellis, T. W. Berger et al., “Nonlinear modeling of causal
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[24] T. F. Coleman, and Y. Li, “On the convergence of interior-reflective Newton
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[26] D. Song, R. H. M. Chan, V. Z. Marmarelis et al., “Nonlinear dynamic modeling
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[27] D. Song, Z. Wang, V. Z. Marmarelis et al., “Parametric and non-parametric
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68

CHAPTER 3: DYNAMIC THRESHSHOLD MODEL AND ITS INTEGRATION
WITH THE SINGLE NEURON MODEL

3.1. Short introduction
Neuron receives presynaptic action potentials (APs) and transforms them to
postsynaptic APs. AP is a rapid membrane potential change caused by the flow of ions
through ion channels in neuron membrane. AP is an all-or-none event and the end
electrical signal formed by a neuron that propagates down the axon to the next stage.
Since the amplitude of AP contains no or little information, the temporal pattern of APs
(spike-train temporal patterns) plays an essential role in carrying the neural information
[1-5]. Therefore, the characteristic of neuron spike-train to spike-train temporal
transformation is critical to the various functions performed by neurons, just to name a
few, such as learning and memory [6, 7]. Unraveling this neuron spike-train to spike-train
temporal transformation is necessary to understand the information processing done by
nervous systems.
Neuron threshold is a voltage level of the membrane potential above which APs
are generated. Variation to the threshold significantly affects neuron spike-train to spike-
train temporal transformation [Azouz, find more references]. Nevertheless, in most
neuron models which have an explicit threshold term (non-compartmental model),
threshold is usually assumed to be a constant [7-12]. However, evidence shows that
threshold is not constant but nonlinear dynamical influenced by the AP firing history [13-
16]. In this chapter, I aim to study the threshold dynamics according to the AP firing
69

history. To do so, there are four major concerns: 1) the data collection of physiologically
plausible intracellular activity, 2) a method of consistent threshold measurement, 3) a
data-driven methodology to capture threshold dynamics according to the AP firing
history with minimum assumptions, and 4) the comparison of spike prediction
performance made by dynamical threshold and constant threshold.
The spike-train temporal transformation between neurons is a complex process
that involves calcium-influx in presynaptic terminal, presynaptic vesicle release of
neurotransmitter, postsynaptic transduction, synaptic integration, somatic integration, AP
generation, AP back-propagation into the dendritic arbors, and retrograde signaling
towards the presynaptic terminal [17, 18]. All these mechanisms are underlying
mechanisms affecting the variation of neuron threshold, and these mechanisms
intervolvingly contribute to the characteristics of neuron spike-train to spike-train
transformation. In order to collect intracellular activity that includes the most
mechanisms, broadband stimulation trains of single all-or-none pulses were delivered to
the synaptic region of CA1 stratum radiatum containing Shaffer collaterals. The all-or-
none stimulation pulses mimic APs. The inter-spike-intervals (ISIs) of the stimulation
trains follow a broadband distribution. These stimuli can elicit a broad range of
physiologically plausible responses and nonlinearities resulted from the interactions of
the mechanisms mentioned above [3, 7, 19].
In order to study threshold dynamics, I need an algorithm to consistently measure
neuron threshold under high-order system conditions. The physiological definition of
neuron threshold is straight forward: a voltage level of membrane potential above which
70

APs are generated; however, to obtain this voltage level from a continuous intracellular
tracing is not straight forward. I proposed a two-step methodology: 1) measure AP
turning points by using third-order derivative analysis, and 2) optimize the constant offset
between the turning points and the thresholds by validating spike prediction accuracy (see
more detail in Measuring Threshold section. Since the relationship between AP turning
point and threshold is a constant (linear), the variation to the AP turning point according
to the AP firing history represents the nonlinearities of threshold dynamics (but not the
linearity). To perform accurate spike predictions, the estimation of the constant offset
between AP turning point and threshold is necessary.
In this chapter, I aim to develop a data-driven nonlinear input-output model to
capture threshold dynamics according to the AP firing history. An up to third-order
Volterra model was developed. The model takes AP firing history as input and threshold
value as output (see Fig. 3.1). The model is estimated using collected patch-clamp
recordings. All open parameters are optimized by minimizing rigorously defined error
terms. In other words, no arbitrary parameter manipulations are involved. This dynamic
threshold model requires minimum assumptions, and avoids errors resulted from
incomplete or biased knowledge [5, 20, 21].
71

Fig. 3.1. The dynamic threshold model. The model takes AP firing history as input and threshold value as
output.

The dynamic threshold model was further combined with a previously developed
single neuron model to replace its original constant threshold term. The spike prediction
performances made by constant and dynamic threshold were compared and dynamic
threshold showed 33% improvement. This improvement confirms the significance of
threshold dynamics to neuron spike-train to spike-train temporal transformation. Hence,
this threshold model exemplifies the benefit of using data-driven nonlinear dynamical
input-output model to decompose underlying neurophysiological signals and to provide
implications for further mechanism studies.
72

3.2. Measuring Threshold
3.2.1. Action potential (AP) turning point measurement
The AP turning point is where neuron membrane potential starts drastically
increasing (see Fig. 3.2), and is usually assumed to be threshold. However, I show that
there is a constant offset between the turning point and threshold (see Fig. 3.3). Therefore,
I proposed a two-step methodology to capture threshold from intracellular recordings: 1)
measure AP turning points by using third-order derivative analysis, and 2) optimize the
constant offset between the turning points and the thresholds by validating spike
prediction accuracy.
The mathematical measurement of the AP turning point can be done in different
ways [13, 22]. I utilized the third-order derivative analysis method suggested by Henze
and Buzsáki in 2001 [13]. The analysis suggested that the timing of the AP turning point
is the same as the first peak of its third-order derivative (see Fig. 3.2), which is calculated
as follows,
3
32 1 1 2 3
33
813 13 8
8
tt t t t t t
dV V V V V V V
dt t
    
   


(3.1)
In (3.1), V represents membrane potential (mV) and t represents time (ms). This equation
is derived from Taylor series expansions to be accurate to O( ∆t
4
) [23].

73

Fig. 3.2. The third-order derivative analysis of AP turning point. The solid line is the recorded membrane
potential of an AP. The dashed line is its third-order derivative. Arrows indicate the AP turning point which
happens at the same time as the first peak of the third-order derivative.

To verify whether the third-order derivative analysis provides consistent
measurement under high-order system conditions and to study the relationship between
AP turning point and threshold, I applied four stimulation patterns which have various
physiological and modeling significances: 1) single-pulse (Fig. 3.3A), 2) pair-pulse
stimulation with a 25 ms ISI (Fig. 3.3B), 3) pair-pulse stimulation with a 90 ms ISI (Fig.
3.3C), 4) triple-pulse stimulation with 90 and 25 ms ISIs in which an AP was evoked by
the second pulse (Fig. 3.3D). In each pattern, the last stimulation had increasing
intensities from 25 to 800 A, and its responses, either the peaks of PSPs or turning
74

points of APs, were analyzed in the right column of Fig. 3. The peaks of PSPs elicited by
certain stimulation intensities at which APs are also evoked represents neuron threshold.
In Fig. 3.3, four stimulation patterns with various physiological and modeling
significances were applied: (A) single-pulse, (B) pair-pulse with a 25 ms ISI, (C) pair-
pulse with a 90 ms ISI, and (D) triple-pulse with 90 and 25 ms ISIs. All data in Fig. 3 are
from a single representative neuron, and are normalized using the average threshold (blue
dash line) value observed in (A), indicated by the black triangle ( ◄). The four panels on
the left column are overlapping PSP and AP intracellular recordings. In each panel, one
representative PSP and AP traces are highlighted with dark lines. Arrows ( ↑) indicate
presynaptic stimulation. The star symbol ( ) indicates the last stimulus in each pattern,
which has increasing intensities from 25 to 800 A. The analysis of the last stimulus is
plotted in the right column panels, which is either the peak of a PSP (blue circle) or the
turning point of an AP (red circle). The blue dashed line is the average of PSP peak
amplitudes under the intensities to which APs are also evoked; this line also represents
the threshold. The red dashed line is the average of all AP turning points.
75

Fig. 3.3. AP turning point and threshold analyses.
76

3.2.2. The constant offset between AP turning point and threshold
In Fig. 3.3, four stimulation patterns with various physiological and modeling
significances were applied to 1) test whether the third-order derivative analysis can
provide consistent measurement under high-order system conditions, 2) study the
relationship between AP turning point and neuron threshold. In Fig. 3.3A, a single-pulse
stimulation was applied. This pattern elicits the first-order PSP dynamics (no preceding
presynaptic stimulation), and first-order threshold dynamics (no preceding AP). The
average normalized AP turning point is 109.8%; the average normalized threshold is
100%; yielding 9.8% difference. In Fig. 3.3B, a pair-pulse stimulation with a 25 ms ISI
was applied. This pattern elicits the second-order PSP dynamics (one preceding
presynaptic stimulation), and first-order threshold dynamics (no preceding AP). Hence,
25 ms is the peak time of a regular PSP response. The average normalized AP turning
point is 109.4%; the average normalized threshold is 97%; yielding 12.4% difference. In
Fig. 3.3C, a pair-pulse stimulation with a 90 ms ISI was applied. This pattern elicits the
second-order PSP dynamics (one preceding presynaptic stimulation), and first-order
threshold dynamics (no preceding AP). Hence, 90 ms is the time where the strongest
second-order PSP facilitation happens (shown previously) [24]. The average normalized
AP turning point is 108.1%; the average normalized threshold is 97.3%; yielding 10.8%
difference. In Fig. 3D, a triple-pulse stimulation with 90 and 25 ms ISIs was applied in
which AP was evoked by the second stimulus. This pattern elicits the third-order PSP
dynamics (two preceding presynaptic stimulation), and second-order threshold dynamics
77

(one preceding AP). The average normalized AP turning point is 135%; the average
normalized threshold is 122.5%; yielding 12.5% difference.
Fig. 3.3 shows that the third-order derivative analysis suggested by Henze
provides consistent measurement of AP turning points under high-order system
conditions. In other words, this analysis measures AP turning points accurately in
continuous intracellular recordings to which broadband stimulation conditions are
imposed. Also, Fig. 3.3 shows that there is a relatively 10% constant offset between AP
turning points and thresholds (9.8~12.5% differences across all four patterns). Across all
collected cells, AP turning points are always larger than thresholds, but the actual offset
value is different to each cell and needs to be optimized individually. This constant
relationship has two implications: 1) the variation to the AP turning points represents the
nonlinearities of threshold dynamics (but not the linearity), and 2) the offset between the
AP turning point and threshold needs to be optimized individually to perform accurate
spike prediction.
78

3.3. Modeling Threshold Dynamics
3.3.1. Estimation of the dynamic threshold model
In Fig. 1, yh, the input spike-trains of the dynamic threshold model, can be
expressed as follows,
1
() ( )
N
i
i
yh t t t 



(3.2)
in which N is the total number of APs and t
i
is the time to the i-th AP. The threshold
value , the output of the dynamic threshold model, can be expressed as follows [5, 25],
1
33 2
12 3 12
12
12
111
() ( ) ( ) () ( , ) ( ) ( )
j LL
jjj
jjj
tc f c jv t c jjv tv t
 
 


  

(3.3)
in which, L is the total number of Laguerre basis functions; c

are Laguerre coefficients
estimated with least square estimations as in (3.5); f is the constant offset between the AP
turning point and the threshold found by minimizing spike prediction error rate (SPER)
defined in (3.7) using try-and-error method (see Fig. 3.4A); and ()
j
vn

are the
convolutions of Laguerre basis functions and yh expressed as follows,
1 ( )
() ( )
i
N
jji
itM tt
vt b t t


 


(3.4)
in which,
j
b

represents Laguerre basis functions; M

is a memory window sufficiently
longer than threshold dynamics.
The least square estimation process for Laguerre coefficient estimation is
expressed as follows,
79




1
2
2
3
3
T1T
(1 )
1
(1 ) 1
(1)
(1)
(2)
() ()
(1)
()
()
pq N
N
pq
c
c
cp VV V
c
N
cq






 

 



 


 


 
  
  

 

 

  






(3.5)
In (3.5), 

represents the measured AP turning points; p is the total number of second-
order convolutions
2
1
j
N
V




; q is the total number of third-order convolutions
3
1
j
N
V


 
 
;
V is the concatenated matrix of all
j
V

expressed as,
33 2
11 2 2
(1 ) (1 ) (1 )
1 , ,
jjL j jL j jL
VV V V
 
    
 


(3.6)
Spike prediction error rate (SPER) is expressed as follows,
Number of False-Positive + Number of False-Negative
SPER=
Total Number of Stimulations
(3.7)
Threshold value is generally higher with shorter ISI as shown in Fig. 3.4B. Once the
model is estimated, the model can predict the threshold values to each presynaptic input
stimulation x (see Fig. 3.6), which is expressed as,
1
() ( )
N
i
i
xttt 



(3.8)
A sample trial of threshold prediction made by the dynamic threshold model is shown in
Fig 3.4C.

80

Fig. 3.4. Estimation of dynamic threshold model. A: Estimation of the offset f by using try-and-error
method to minimize SPER. B: Measured threshold vs. ISI. It shows the general phenomena that shorter ISI
is correlated with higher threshold. C: Sample trial of threshold prediction made by the dynamic threshold
model.

3.3.2. Reconstructions of Volterra kernels
Volterra kernels were reconstructed with the Laguerre coefficients c

and the
constant offset f as follows,
1
1
kc f



(3.9)
81

2
11 1
11
,2 3 1 1 1 1 1 1
11
( , , ) ( , , )() ( )
i
ii
i
j
L
ii i i j j i
jj
km m c j jbm bm





   



  
(3.10)
where, m are the elapses of time.
To examine the nonlinear effect to threshold resulted from a certain number of
preceding APs, I utilized the notion of response function (r) [5, 20, 21, 24]. Response
functions (r) can be calculated from Volterra kernels (k) as follows,
11
rk


(3.11)
22 3
() ( , ) rk m k mm

 (3.12)
31 2 3 1 2
(, )2(, ) rm m k m m

 (3.13)
In (3.11), the first-order response function (r
1
) is a scalar (see Fig. 3.5A), representing the
expected threshold value of current input (presynaptic stimulation) when there is no
preceding AP within the memory window. In (3.12), the second-order response function
(r
2
) is a curve (see Fig. 3.5B), representing the nonlinear effect to the threshold of current
input resulted from each preceding single APs with m ISI. In (3.13), the third-order
response function (r
3
) is a surface (see Fig. 3.5C), representing the nonlinear effect to the
threshold of current input resulted from each preceding pairs of APs with m
1
and m
2
ISIs.

3.3.3. Response functions
A set of representative response functions (r
1
, r
2
, and r
3
) are shown in Fig. 3.5.
The first-order response function r
1
is 8.9 mV. The second-order response function r
2

starts from the highest positive peak 5.1 mV, decays to the first valley 0.26 mV at 210
82

ms, rises to the second peak 0.71 mV at 445 ms, and then slowly decays lower than 0.1
mV at 1200 ms. The third-order response function r
3
starts from -0.09 mV, symmetrically
decays to the first valley -1.1 mV at (40 ms, 160 ms) and (160 ms, 40 ms), rises to the
highest peak 0.31 mV at (160 ms, 160 ms), decays to the second valley -0.4 mV at (400
ms, 400 ms), and then slowly rises back to -0.01 mV at (800 ms, 800 ms).

83

Fig. 3.5. Threshold dynamics response functions. First-order response function (r
1
). Second-order response
functions (r
2
). Third-order response functions (r
3
).
84

3.4. Integration of the Dynamic Threshold Model with the Single Neuron Model
3.4.1. Spike prediction accuracy improvements
A single neuron model with a constant threshold which captures neuron sub- and
supra-threshold nonlinear dynamics was previously developed (see Fig. 3.6A) [24]. The
constant threshold was optimized by minimizing SPER. The dynamic threshold model
was incorporated into this single neuron model to replace the constant threshold
component (see Fig. 3.6B). And, out-of-sample spike prediction accuracies evaluated
using SPER performed by constant threshold and dynamic threshold were compared (see
Fig. 3.7 and 3.8). In both Fig. 3.6A and 3.6B, x represents presynaptic stimulation train;
w represents the pre-threshold (non-spiking) membrane potential, which is the summation
of u (PSP) and a (spike-triggered after-potential); y represents the overall somatic PSP
and AP voltage trace; yh represents all-or-none AP train. See more mathematical
expressions of the single neuron model in Chapter 2.

85

Fig. 3.6. Model structure of the single neuron model with the constant threshold (A) and the dynamic
threshold (B).

Two representative out-of-sample prediction clips from a single neuron made with
constant threshold and dynamic threshold are shown in Fig. 3.7A and 3.7B. Fig. 3.7A
shows predictions in which constant threshold accurately predicted the occurrences of
APs; dynamic threshold worked just as well. Fig 3.7B shows predictions in which
86

constant threshold made thee false-positive predictions; however, dynamic threshold
avoided two of them and significantly improved the spike prediction accuracy.
87

Fig. 3.7. Representative sample clips of spike predictions made with constant threshold and dynamic
threshold. In both A and B, dashed lines in the middle panel indicate the optimized constant threshold;
black dots in the bottom panels indicate threshold values predicted by the dynamic threshold model. In the
case when constant threshold worked perfectly, dynamic threshold worked just as well (A). However, in
the case when constant threshold made false-positive errors, dynamic threshold significantly improved the
prediction accuracy (B).
88

Fig. 3.8A shows the false-positive and true-positive spike prediction performance
made by all possible constant threshold (the black line, also called receiver operating
characteristic (ROC) curve analysis) and dynamic threshold (the black dot). The gray
circle indicated the performance made by the optimal constant threshold. It shows that the
performance achieved by dynamic threshold is impossible to be achieved by any constant
threshold value. Histogram of SPER made by optimized constant threshold and dynamic
threshold is shown in Fig. 3.8B. The mean SPER made by constant threshold over all
datasets is 24.8%, and dynamic threshold is 14.7%. The average SPER improvement
made by the dynamic threshold model within each single prediction trials comparing with
optimized constant threshold is 33% and the histogram is shown in Fig. 3.8C.
89

Fig. 3.8. A: ROC curve analysis. The gray circle represents the spike prediction performance made by
optimal constant threshold. The black dot represents the spike prediction performance made by the dynamic
threshold. B: Histogram of SPER made by optimized constant threshold and dynamic threshold. C:
Histogram of SPER improvement made by the dynamic threshold model comparing to optimized constant
threshold.
90

3.5. Discussion
In this chapter, I developed a data-driven high-order nonlinear model to capture
threshold dynamics according to the AP firing history. All model parameters of this
dynamic threshold model are simultaneously constrained using intracellular recordings
by minimizing rigorously defined error terms. This data-driven property provides three
modeling significances [20, 21, 24, 26]: 1) no arbitrary parameter manipulations involves
in the model estimation, 2) minimum assumptions were made to the model, and 3)
modeling errors due to biased knowledge or unknown mechanisms are avoided. This
dynamic threshold model has input (AP firing history) and output (threshold value) that
are common to all spiking neurons, meaning the model can be applied to all kinds of
spike-input, spike-output neurons. This model represents a tool that analyzes and presents
threshold dynamics in a quantifiable manner. Finally, this model is computationally
efficient. The model contains only 11 open parameters and can be easily estimated using
a regular PC (AMD Phenom 9750).
The dynamic threshold model was integrated into a single neuron model (Fig.
3.6B) and showed 33% spike prediction improvements comparing to its original constant
threshold (Fig. 3.6A). This improvement shows that certain features of the threshold
dynamics were not captured originally. In other words, the dynamic threshold model
further decomposes underlying physiological mechanisms and improves model
accuracies at the same time. This enhancement of both physiological interpretation and
model predictability demonstrates the benefits of a hybridized neuron model structure by
91

combining both mechanistic (parametric) and input-output (nonparametric) components.
This is to say configuring input-output modeling components (as in Fig. 3.6B, threshold ,
feedforward k, and feedback h) according to the common principles of neuron signal flow
(respectively as in Fig. 3.6B, spike generation, presynaptic spike to PSP transformation,
spike-triggered after-potential).
Configuration of a hybrid model to decompose underlying mechanisms needs to
be in accordance to the nature of data collection paradigms. In 2007, Song and et al.
developed a similar approach to capture neuron spike-train to spike-train transformation.
That approach demonstrated the feasibility of using extracellularly recorded unitary
activities (spikes) as both model input and output [7, 27]. Since extracellular spike-train
recordings do not provide direct information of neuron membrane potential as oppose to
intracellular recordings, measuring threshold from the recorded data cannot be performed.
Therefore, estimation of a dynamic threshold model as reported in this chapter is not
possible. However, since the threshold dynamics is spike-dependent, threshold dynamics
along with other spike-dependent mechanisms are lumped and captured altogether by the
feedback kernel in Song's model. This means the nonlinear effects resulted from
threshold dynamics are absorbed by the feedback kernels in Song's model. This
characteristic also demonstrates the benefit of using data-driven input-output
methodology to avoid modeling errors due to unknown mechanisms.
The dynamic threshold model provides implications for further mechanism
studies. In Fig. 6, the double-positive-peak curve of the second-order response function
and double-negative-peak surface of the third-order response function suggest at least
92

two or more major ion channels with different re-activation/de-activation time constants
interactively contribute to the threshold dynamics [13, 14, 28-33]. A detail
compartmental modeling is needed to isolate the contributions from different ion
channels. The data suggest that there is a relatively constant delay (either in time or in
potential) between the very moment when sodium channel entering the positive-feedback
phase (AP initiation) and the timing of drastic increase in membrane potential, which
resulted in the constant offset between the AP turning point and threshold (see Fig. 3.3).
A compartmental model is also needed to confirm this point.
93

3.6. References

[1] T. W. Berger, G. Chauvet, and R. J. Sclabassi, “A biologically based model of
functional properties of the hippocampus,” Neural Netw, vol. 7, no. 6/7, pp. 1031-
1064, 1994.

[2] R. J. Sclabassi, J. L. Eriksson, R. L. Port et al., “Nonlinear systems analysis of the
hippocampal perforant path-dentate projection. I. Theoretical and interpretational
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[3] T. W. Berger, J. L. Eriksson, D. A. Ciarolla et al., “Nonlinear systems analysis of
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[4] T. W. Berger, J. L. Eriksson, D. A. Ciarolla et al., “Nonlinear systems analysis of
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[5] V. Z. Marmarelis, Nonlinear Dynamic Modeling of Physiological Systems: Wiley-
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[6] T. W. Berger, A. Ahuja, S. H. Courellis et al., “Restoring lost cognitive function,”
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[7] D. Song, R. H. M. Chan, V. Z. Marmarelis et al., “Nonlinear dynamic modeling
of spike train transformations for hippocampal-cortical prostheses,” IEEE
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[8] C. D. Geisler, and J. M. Goldberg, “A stochastic model of the repetitive activity
of neurons,” Biophys J, vol. 6, no. 1, pp. 53-69, Jan, 1966.

[9] W. Gerstner, and W. Kistler, Spiking neuron models: single neurons, populations,
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[10] E. M. Izhikevich, “Simple model of spiking neurons,” IEEE Trans Neural Netw,
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[11] C. Koch, and I. Segev, “The role of single neurons in information processing,”
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94

[12] C. Koch, Biophysics of Computation: Information Processing in Single Neurons:
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[13] D. A. Henze, and G. Buzsaki, “Action potential threshold of hippocampal
pyramidal cells in vivo is increased by recent spiking activity,” Neuroscience, vol.
105, no. 1, pp. 121-30, 2001.

[14] R. Azouz, and C. M. Gray, “Cellular mechanisms contributing to response
variability of cortical neurons in vivo,” J Neurosci, vol. 19, no. 6, pp. 2209-23,
Mar 15, 1999.

[15] R. Azouz, and C. M. Gray, “Dynamic spike threshold reveals a mechanism for
synaptic coincidence detection in cortical neurons in vivo,” Proc Natl Acad Sci U
S A, vol. 97, no. 14, pp. 8110-5, Jul 5, 2000.

[16] M. J. Chacron, B. Lindner, and A. Longtin, “Threshold fatigue and information
transfer,” J Comput Neurosci, vol. 23, no. 3, pp. 301-11, Dec, 2007.

[17] B. Hille, Ion Channels of Excitable Membranes, 3 ed.: Sinauer, 2001.

[18] D. Johnston, and S. M. S. Wu, Foundations of cellular neurophysiology: The MIT
Press, 1995.

[19] G. Gholmieh, S. Courellis, V. Marmarelis et al., “An efficient method for
studying short-term plasticity with random impulse train stimuli,” J Neurosci
Methods, vol. 121, no. 2, pp. 111-27, Dec 15, 2002.

[20] D. Song, V. Z. Marmarelis, and T. W. Berger, "Parametric and non-parametric
models of short-term plasticity." pp. 1964-1965.

[21] D. Song, V. Z. Marmarelis, and T. W. Berger, “Parametric and non-parametric
modeling of short-term synaptic plasticity. Part I: Computational study,” J
Comput Neurosci, vol. 26, no. 1, pp. 1-19, Feb, 2009.

[22] M. Sekerli, C. A. Del Negro, R. H. Lee et al., “Estimating action potential
thresholds from neuronal time-series: New metrics and evaluation of
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1665-1672, Sep, 2004.

[23] B. Raton, Standard Mathematical Tables and Formulae, 29th ed.: CRC, 1991.

[24] U. Lu, D. Song, and T. W. Berger, “Nonlinear dynamic modeling of synaptically
driven single hippocampal neuron intracellular activity,” IEEE Transactions on
Biomedical Engineering, 2011.
95

[25] V. Z. Marmarelis, “Identification of nonlinear biological systems using Laguerre
expansions of kernels,” Ann Biomed Eng, vol. 21, no. 6, pp. 573-89, Nov-Dec,
1993.

[26] V. Z. Marmarelis, and T. W. Berger, “General methodology for nonlinear
modeling of neural systems with Poisson point-process inputs,” Math Biosci, vol.
196, no. 1, pp. 1-13, Jul, 2005.

[27] D. Song, R. H. Chan, V. Z. Marmarelis et al., “Nonlinear modeling of neural
population dynamics for hippocampal prostheses,” Neural Netw, vol. 22, no. 9, pp.
1340-51, Nov, 2009.

[28] H. Y. Jung, T. Mickus, and N. Spruston, “Prolonged sodium channel inactivation
contributes to dendritic action potential attenuation in hippocampal pyramidal
neurons,” J Neurosci, vol. 17, no. 17, pp. 6639-46, Sep 1, 1997.

[29] T. Mickus, H. Jung, and N. Spruston, “Properties of slow, cumulative sodium
channel inactivation in rat hippocampal CA1 pyramidal neurons,” Biophys J, vol.
76, no. 2, pp. 846-60, Feb, 1999.

[30] H. Murakoshi, and J. S. Trimmer, “Identification of the Kv2.1 K+ channel as a
major component of the delayed rectifier K+ current in rat hippocampal neurons,”
J Neurosci, vol. 19, no. 5, pp. 1728-35, Mar 1, 1999.

[31] M. C. Sanguinetti, and N. K. Jurkiewicz, “Two components of cardiac delayed
rectifier K+ current. Differential sensitivity to block by class III antiarrhythmic
agents,” J Gen Physiol, vol. 96, no. 1, pp. 195-215, Jul, 1990.

[32] N. Spruston, Y. Schiller, G. Stuart et al., “Activity-dependent action potential
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[33] M. C. Spruston N., Structure and function properties of hippocampal neurons:
Oxford University Press, 2007.

96

CHAPTER 4: NONLINEAR DYNAMICAL MODELING OF LONG-TERM
SYNAPTIC PLASTICITY

4.1. Introduction
Single neurons, serving as basic cellular units of every neural network, receive
presynaptic spike-trains and transform them into postsynaptic spike-trains. As argued in
Chapter 3, all functions performed by neurons are encoded in this spike-train to spike-
train transformations. Learning and memory, among all cognitive functions, have
especially attracted much more attention among neuroscientists. Long-term potentiation
(LTP), a long lasting potentiation in synaptic transmissions, is considered to be an
important cellular mechanism mediating learning and memory since its discovery in 1966
by Terje Lømo. Understanding the nonlinear changes in the spike-train to spike-train
transformation caused by LTP is fundamental to unravel how learning and memory work
[1].
LTP is a wide-spread phenomenon possibly expresses at every excitatory synapse
in the brain. LTPs at different synapses have different expressing mechanisms. Even
though LTP has been studied for a long time with enormous efforts, the understanding of
it has been limited to the concept of synaptic strengthening. However, synaptic
strengthening addressed merely the first-order response function effect (single-pulse
effect) in kernel analysis and left higher-order effects (paired-pulse, triple-pulse, and etc.)
unanswered. This limited concept of LTP is potentially due to biased theoretical setting
and non-conclusive experimental procedures.
97

LTP should be viewed as an important mechanism that changes the encoding
ability of a neuron. This encoding ability is represented in the neuron spike-train to spike-
train temporal pattern transformation. As described in previous chapters, the kernel
analysis developed in this thesis can analyze and quantify this spike-train transformation.
Berger and Sclabassi (1988) reported kernel analyses of pre- and post-LTP dynamic
changes [2]. The study was conducted using extracellular recording representing
saturated synaptic responses of a population of neurons. In this thesis, I like to extend this
idea to a single neuron and unsaturated synaptic response level.
LTP research during the 1990s was enlivened by intense debate as to whether
NMDAR-dependent LTP is expressed presynaptically or postsynaptically. Over the
years, considerable evidence has accumulated in favor of both. Presynaptic mechanism
focused on the enhancement of vesicle release mechanisms [3]. LTP might increase the
presynaptic calcium level, which increases the vesicle release probability [4-12], which
increases the amplitudes of single isolated EPSPs. The increased vesicle release
probability results in a less ready-to-release pool of vesicles in the case of repetitive
stimulations which leads to depressions in higher order kernels. This is known as vesicle
depletion theory.
Large and compelling set of data that gives detailed insights into how
postsynaptic changes are brought about by posttranslational modifications of existing
receptors and trafficking of receptors between the cytoplasm and extra-synaptic and
synaptic membrane. Abundant evidence shows that LTP enhances postsynaptic current
by increasing either the redistribution of AMPAR [13-15], sensitivity of AMPAR [16-
98

19], the total number of AMPAR [14, 20-22], or modifications of postsynaptic voltage-
dependent ion channels [23]. The enhanced synaptic current may change the nonlinear
dynamics of postsynaptic voltage-dependent responses. The changes in those voltage-
dependent ion channel dynamics may interactively result in facilitations of linear
responses and depressions of high-order nonlinear responses.
In this chapter, we developed a two-stage cascade model to describe the nonlinear
dynamics of both pre- and post-synaptic mechanisms (Fig. 4.1). In order to collect the
requisite information-rich input-output datasets, the recorded neurons were stimulated
pre-synaptically using broadband random interval trains (RITs). The stimulations are all-
or-none electrical pulses mimicking action potentials. The broadband RITs elicit
biologically plausible processes in both pre- and post-synaptic regions. Whole-cell
patch-clamp was utilized to record excitatory post-synaptic currents (EPSCs) and
excitatory post-synaptic potentials (EPSPs).

Fig 4.1. Model structure of the two-stage cascade model.
99

Both stages of the model are developed using third-order Volterra kernels to
describe the transformation from the input to the output. The first stage of the model
represents pre-synaptic mechanisms and describes the nonlinear dynamical
transformation from pre-synaptic spike trains to transmitter vesicle release strengths. The
second stage of the model represents post-synaptic mechanisms and describes the
nonlinear dynamical transformation from vesicle release strengths to EPSPs.
To conduct this research, it is critical to estimate vesicle release strengths from the
recorded EPSCs [24, 25]. EPSC is a mixture of the nonlinear dynamics of both pre-
synaptic vesicle release mechanism and post-synaptic glutamate-receptor channel
mediated current dynamics. In this study, the isolation of the vesicle release mechanism
is achieved by deconvolving EPSC recordings with an averaged isolated EPSC waveform
observed in the same EPSC recordings (see more details in the next section). This
operation is based on the linearity of AMPA receptor channel-mediated (AMPA
RC
-
mediated) current dynamics under three experimental conditions: 1) the post-synaptic
current responses (i.e., EPSCs) are recorded under voltage-clamp at resting potential, 2)
the stimulation intensity is sufficiently low, so that the Mg
2+
cannot be expelled from the
NMDA receptor channel (NMDA
RC
), and 3) the smallest inter-spike interval in the RITs
applied is 10 ms, so that the nonlinearity due to AMPA
RC
desensitization is reduced.

100

4.2. Materials and methods
4.2.1. Electrophysiology
Hippocampal slices (400 m thick) were prepared from young adult Sprague-
Dawley rats (male, four-week-old) with standard procedures in iced cutting (sucrose)
solution. The connection between CA3 and CA1 was surgically disrupted to reduce
spontaneous activities in CA1 neurons. Slices were maintained and recorded in artificial
cerebral spine fluid (ACSF) at room temperature containing (in mM): NaCl 124, KCl 2.5,
NaH
2
PO
4
1.25, NaHCO
3
26, Glucose 10, MgSO
4
1, Ascorbic Acid 2, and CaCl
2
2; at pH
7.4 and 295 mOsmol. The perfusion ACSF medium for recordings contained 20uM
Picrotoxin.

-150
-100
-50
0
EPSC
1 1.5 2 2.5 3 3.5 4 4.5
0
2
4
6
8
Time (s)
EPSP
pA
mV

Fig. 4.2. Representative recordings of EPSCs and EPSPs evoked by the same pre-synaptic RIT stimulation
pattern.

101

A bipolar stimulating electrode was placed on slices based on visual cues as to
activate Schaffer collaterals in the CA1 stratum radiatum region. Recording micro-
pipette electrodes with a 4 M Ω tip resistance were prepared using a standard puller. The
internal solution of the recording electrode contained (in mM): K-SO
3
135, HEPES 10,
Mg-ATP 2, and Na
3
-GTP 0.25; at pH 7.3 and 290 mOsmol. Random-interval EPSCs and
EPSPs were consistently evoked using Poisson distributed RITs with a 2 Hz mean
frequency (inter-spike interval 10~4500 ms) and recorded either in voltage-clamp or
current-clamp mode (see Fig. 4.2). The stimulation intensity was set as to evoke a 3 mV
peak amplitude in an isolated EPSP.

4.2.2. Estimating vesicle release strengths from the recorded random-internal EPSCs

Fig. 4.3. The transformation from pre-synaptic spikes to EPSCs involves pre-synaptic mechanisms and
AMPA
RC
-mediated current dynamics.

The presynaptic RITs (action potentials), x(t), is expressed in a series of impulse
responses as follows,
102

1
() ( )
N
i
i
xttt 



(4.1)
where N represents the total number of action potentials and t
i
represents the timing of the
i-th action potential. The transmitter vesicle release strength, ys(t), is expressed in a
series of impulse responses with varying amplitudes as follows,

1
() ( )
N
ii
i
ys t A t t 



(4.2)
where A
i
represents the release strength in response to the i-th pre-synaptic action
potential.
Here, we make a reasonable assumption that, with the experimental conditions
applied in this study, the AMPA
RC
-mediated current dynamics is a linear process. Thus,
the relationship between EPSC, yc(t), and release strength, ys(t), is expressed as follows,
() ( ) ( ) yc t hc ys t d   

(4.3)
where hc( ) is the linear impulse response of the transformation between ys(t) and yc(t),
obtained by averaging all isolated EPSC responses in the same recording. Isolated EPSC
is defined as an EPSC having 3000 ms silence period (no stimulation) before and no
overlapping EPSC after it (i.e., the first-order EPSC response). With (4.3), ys(t) can be
obtained by deconvolving yc(t) using hc(  .

4.2.3. Volterra modeling of pre- and post-synaptic mechanisms
In Figure 1, the transmitter vesicle release strengths, ys(t), is expressed with
103

Volterra series as follows,
1
12
12
12
1
31 2
11
() ( ) ()
(, ) () ()
L
s
ii j
j
j L
ss
jj
jj
ys t A c c j v t
cj j v tv t


 



(4.4)
where L denotes the number of Laguerre basis functions, c denote Laguerre coefficients,
and
s
j
v
denote the convolutions of Laguerre basis functions and x(t), and is expressed as,

() ( )
i
s
j ji
tM t t
vt b t t
 

 (4.5)
where b denotes Laguerre basis functions and M denotes the memory window.
The EPSP, yp(t), is expressed as follows,
1
12
12
12
12 3
12 3
01
1
21 2
11
31 2 3
11 1
() ( ) ()
(, ) () ()
( , , ) () () ()
L
p
j
j
j L
pp
jj
jj
jj L
pp p
jj j
jj j
yp t c c j v t
cj j v tv t
cj j j vtv tv t


 





 
(4.6)
where c are Laguerre parameters and
p
j
v
are convolutions of ys(t) and Laguerre basis
functions, ()
j
bt , expressed as follows,
0
() ( ) ( )
M
p
jj
vt b yst

 



(4.7)

All c in (4.4) and (4.6) are estimated using the least-squares method [26].
104

4.2.4. Reconstructions of Volterra kernels and response functions
Volterra kernels of the pre-synaptic mechanism model (first stage) are
reconstructed with the estimated Laguerre coefficients c in (4.4) as follows,
11
kc  (4.8)
11
11
,2 3 1 1 1 1 1
11
( ,..., ) ... ( ,..., ) ( )... ( )
i
i
LL
ii i i i j j i
jj
km m cj jbm bm


  


 (4.9)

where m represents the inter-spike interval (ISI). And k
1
=c
1
is a scalar. Volterra kernels
of the post-synaptic mechanism model (second stage) are reconstructed with the
estimated Laguerre coefficients c in (4.6) as follows,
1
1
,1 3 1 1 1
11
( ,..., ) ... ( ,..., ) ( )... ( )
ii
i
LL
ii i k i j j i
jj
kcjjbb 



 (4.10)
Response functions can be calculated from the Volterra kernels k to provide
intuitive physiological interpretations in terms of single-pulse, paired-pulse, and triple-
pulse effects [24, 26].

4.2.5. The linearity of AMPA
RC
-mediated current dynamics
By expressing the relationship between yc(t) and ys(t) in a convolution form as
(4.3), we are assuming that the AMPA
RC
-mediated current dynamics is a linear process
with an impulse response being the averaged isolated EPSC hc( ). This is a key
assumption in this report and needs to be verified.
105

1 2 3 4 5 6
-200
-160
-120
-80
-40
0
Time (s)
Current (pA)
Reconstructed EPSCs
Recorded EPSCs
-150 -100 -50 0
-200
0
Reconstructed EPSCs (pA)
Recorded EPSC (pA)
B
A
-150
-100
-50

Fig. 4.4. The reconstructed EPSCs produced by convolving ys(t) and hc(t). Overlapping (A) and scatter plot
(B) of the recorded EPSCs and reconstructed EPSCs.

First, in order to make sure that the EPSCs recorded are AMPA
RC
-mediated
instead of a mixture of AMPA
RC
and NMDA
RC
. Two experimental manipulations were
applied to ensure this: 1) EPSCs were recorded using voltage-clamp at the resting
potential which keeps most NMDA
RC
inactivated, because NMDA
RC
needs about 15 mV
depolarization from the resting potential to expel the Mg
2+
ion from its mouth to be
106

activated; 2) stimulation intensity was set as the peak value of an isolated EPSP is about 3
mV, so that in the case that patch-clamp cannot provide sufficient control over some
distal synapses (the space-clamp issue), NMDA
RC
still remains inactivated. Second, we
need to reduce the potential AMPA
RC
nonlinearity in the recorded EPSCs. The potential
nonlinearity is primarily caused by the desensitization of AMPA
RC
during repetitive
stimulations which has been reported to have a time constant of 10 ms [27]. According
to this, the smallest ISI in the applied RITs is set to 10 ms, so that the most effective time
range (0~10 ms) of AMPA
RC
desensitization is not included. Also, the Schaffer
collateral-CA1 synapses have a relatively low (~30%) release probability in response to
pre-synaptic action potentials [28]. Using electrical stimulation results in only ~10% of
the accessible synapses to be activated twice in a two consecutive stimulation, and thus
results in an even weaker desensitization effect beyond 10 ms ISI [25].
Results in Fig. 4.4 show that the AMPA
RC
-mediated nonlinearity is indeed
negligible in the recorded EPSCs. This is indicated in the almost perfect overlap between
the recorded EPSCs and reconstructed EPSCs (see Fig. 4.4A), where the reconstructed
EPSC is obtained by convolving ys(t) and hc( ). The scatter plots of the recorded and
reconstructed EPSC dots almost perfectly on the diagonal (see Fig. 4.4B). Both figures
validate the assumption that of AMPA
RC
-mediated current dynamics is linear, and
justifies the practice of estimating the release strengths, ys, by deconvolving the EPSC
recording using an averaged isolated EPSC observed in itself.
107

4.3. Results and discussion
A representative clip of EPSC recording before and after LTP is shown in Fig. 4.5.
Fig. 4.5. A representative clip of EPSC recordings before and after LTP.

A representative clip of EPSP recording before and after LTP is shown in Fig. 4.6.

Fig. 4.6. A representative clip of EPSP recordings before and after LTP.

As expected, the responses after LTP-induction are larger in general in both EPSC
and EPSP. In order to study the changes in the nonlinear dynamics, I performed kernel
analyses.
108

Presynaptic model response functions are shown in Fig. 4.7.

Fig. 4.7. Response functions of presynaptic model before and after LTP.
109

The left column of Fig. 4.7 is the response functions of the first stage of the model
(presynaptic model), using presynaptic spike as input and pre-LTP EPSC as output. The
normalized second-order (R2) and third-order response function (R3) represent the
nonlinear dynamical pair-pulse and triple-pulse effect resulting from presynaptic
mechanisms. Similarly, the right column of Fig. 4.7 is the response functions of the first
stage of the model, using presynaptic spike as input and post-LTP EPSC as output. The
normalized second-order (R2) and third-order response function (R3) represent the
nonlinear dynamical pair-pulse and triple-pulse effect resulting from presynaptic
mechanisms.
In the case of presynaptic mechanism model, the result of 1) post-LTP R2 minus
pre-LTP R2, and 2) post-LTP R3 minus pre-LTP R3 are shown in the left column in Fig.
4.9. The difference in R2 is within 6%; and 5% in the case of R3. Paired T-test (p
value=0.05) showed none of the points in R2 change and R3 change is significant.

110

Fig. 4.8. Response function of postsynaptic model before and after LTP.
111

The left column of Fig. 4.8 is the response functions of the second stage of the
model (postsynaptic model), using EPSC amplitude as input and Pre-LTP EPSP as
output. The normalized second-order (R2) and third-order response function (R3)
represent the nonlinear dynamical pair-pulse and triple-pulse effect resulting from
postsynaptic mechanisms. Similarly, the right column of Fig. 4.8 is the response
functions of the second stage of the model (postsynaptic model), using EPSC amplitude
as input and Post-LTP EPSP as output. The normalized second-order (R2) and third-order
response function (R3) represent the nonlinear dynamical pair-pulse and triple-pulse
effect resulting from postsynaptic mechanisms.
The reduce of one dimensionality between Fig. 4.7 and Fig. 4.8 is because the
output of the presynaptic model is scalars (EPSC amplitudes); on the other hand, the
output of postsynaptic model is continuous waveforms (EPSP membrane potential).
In the case of post-synaptic mechanisms model, the result of 1) post-LTP R2
minus pre-LTP R2, and 2) post-LTP R3 minus pre-LTP R3 are shown in the right column
in Fig. 4.9. The difference in R2 is over 50%; and 15% in the case of R3. Paired T-test (p
value=0.05) showed the first 50 ms in R2 change is significant. R3 is not significant.

112

Fig. 4.9. Presynaptic and postsynaptic nonlinearity changes.

113

The kernel analysis shows that after LTP is induced, postsynaptic nonlinearities
are significantly changed; while, presynaptic nonlinearity remain unchanged. This result
suggests that the LTP expression locus is postsynaptic.
114

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Abstract (if available)

Abstract Neuron spike-train to spike-train temporal transformation is very important to the functions of neurons. Neurons receive presynaptic (input) spike-trains and transform them into postsynaptic (output) spike-trains. This input-output transformation is a highly nonlinear dynamic process which depends on complex nonlinear physiological processes. Mathematically capturing and quantifying neuron spike-train to spike-train transformation are important to understand the information processing done by neurons.

Compartmental modeling methodology is to simulate and interpret detail neuron physiological mechanisms/processes. The Hodgkin-Huxley model is the most prominent example in this category. However, model structure/parameter of compartmental modeling is specific to the targeted neuron (or type of neurons) and not applicable to the others, and the modeling result is vulnerable to biased or incomplete knowledge. Hence, the number of open parameters is often large, making it computationally inefficient. Integrate-and-fire neuron model is a computationally efficient methodology that received a lot of attention in the past two decades. It is perfect for large-scale simulation, and provides qualitative neuron characterization. However, it is over simplified and provides no or little mechanistic implications or quantifications. Lastly, input-output modeling methodology, which is applied in this study is another major approach to characterize neuron spike-train transformation. Input-output models are data-driven. This leads to an important property that it avoids modeling errors due to biased or incomplete knowledge. The number of open parameters is limited, making the model relatively computationally efficient. In other words, input-output model provides is well balanced between the common modeling dilemma: accuracy and efficiency.

In my study, the purpose is to build a single neuron model that 1) captures both sub- and supra-threshold dynamics based on neuron intracellular activity, 2) is sufficiently general to be applied to all spike-input, spike-output neurons, 3) is computationally efficient.

A nonlinear dynamical single neuron model was developed using Volterra kernels based on patch-clamp recordings. There were two phases in developing this model. In the first phase, a single neuron model with constant threshold was developed. It consists: 1) feedforward kernels (up to third-order) which transform presynaptic spikes into postsynaptic potentials (PSPs), 2) a constant threshold which represents the spike generation process, and 3) a feedback kernel (first-order) which describes spike-triggered after-potentials. The model was applied to CA1 pyramidal cells as they were electrically stimulated with broadband impulse trains through the Schaffer collaterals. This synaptically driven broadband intracellular activities contains a broad range of nonlinear dynamics resulted from the interactions of underlying mechanisms. The model performances were evaluated separately with respect to: PSP waveforms and the occurrence of spikes. The average normalized mean square error (NMSE) of PSP prediction is 14.4%. The average spike prediction error rate (SPER) is 18.8%.

In the second phase, inspired by literatures, a dynamical model was developed to study threshold nonlinear dynamics according to the action potential (AP) firing history. To develop the model, we measured the turning point of AP by analyzing its third-order derivative. The AP turning point has a constant offset relationship with the threshold. In other words, variation to the AP turning point represents the nonlinearities of threshold dynamics. To perform accurate spike prediction, it requires an additional spike prediction validation to optimize that offset (the linearity). This dynamic threshold model was implemented using up to third-order Volterra kernels constrained by synaptically driven intracellular activity described before. This threshold model was integrated into the single neuron model to replace its original constant threshold and showed 33% SPER improvement.

This single neuron model is a hybrid, combining both mechanistic (parametric) and input-output (non-parametric) components. The principles of neuronal signal generation common to all spike-input, spike-output determine the model structure. On the other hand, the specific properties that are variable from neuron to neuron are captured and quantified with descriptive model parameters, which are directly constrained by intracellular recording data. This hybrid representation of both parametric and nonparametric model components partitions data variance with respect to mechanistic sources and thus imposes physiological definitions to the model components and facilitates the biological interpretations of the parameters.

This single neuron model was further applied to analyze long-term potentiation (LTP) in single neurons. The purpose of this application is to separate and quantify the pre- and post-synaptic mechanisms both before and after LTP induction. The single neuron model is modified to be a two-stage cascade model. The first-stage represents presynaptic mechanisms, taking presynaptic spikes as input and excitatory postsynaptic currents (EPSCs) as output. The second-stages represents postsynaptic mechanisms, taking EPSCs as input and excitatory postsynaptic potentials (EPSPs) as output. Preliminary data shows that LTP intensifies the linear responses and reduces the nonlinearities.

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Creator Lu, Ude (author)

Core Title Nonlinear dynamical modeling of single neurons and its application to analysis of long-term potentiation (LTP)

School Viterbi School of Engineering

Degree Doctor of Philosophy

Degree Program Biomedical Engineering

Publication Date 05/27/2011

Defense Date 05/26/2011

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag electrophysiology,long-term potentiation,neural network,neuron,neuron modeling,nonparametric modeling,OAI-PMH Harvest,Volterra kernels,whole-cell patch-clamp

Language English

Contributor Electronically uploaded by the author (provenance)

Advisor Berger, Theodore W. (committee chair), Baudry, Michel (committee member), D'Argenio, David Z. (committee member), Marmarelis, Vasilis Z. (committee member), Song, Dong (committee member)

Creator Email ude.lu77@gmail.com,ulu@usc.edu

Permanent Link (DOI) https://doi.org/10.25549/usctheses-c127-613625

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Legacy Identifier etd-LuUde-7-0.pdf

Dmrecord 613625

Document Type Dissertation

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Type texts

Source University of Southern California (contributing entity), University of Southern California Dissertations and Theses (collection)

Access Conditions The author retains rights to his/her dissertation, thesis or other graduate work according to U.S. copyright law. Electronic access is being provided by the USC Libraries in agreement with the a...

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