Southern Federal University
A.B.Kogan Research Institute for Neurocybernetics
Laboratory of neuroinformatics of
sensory and motor systems
Introduction to modern methods and
tools for biologically plausible modeling
of neural structures of brain
Part II
Ruben A. Tikidji – Hamburyan
rth@nisms.krinc.ru

Previous lecture in a nutshell
1. There is brain in head of human and animal. We use it for thinking.
2. Brain is researched at different levels. However physiological methods
is constrained. To avoid this limitations mathematical modeling is
widely used.
3. The brain is a huge network of connected cells. Cells are called
neurons, connections - synapses.
4. It is assumed that information processes in neurons take place at
membrane level. These processes are electrical activity of neuron.
5. Neuron electrical activity is based upon potentials generated by
selective channels and difference of ion concentration in- and outside
of cell.
6. Dynamics of membrane potential is defined by change of
conductances of different ion channels.
7. The biological modeling finishes and physico-chemical one begins at
the level of singel ion channel modeling.

Previous lecture in a nutshell
8. Instead of detailed description of each ion channel by energy function
we may use its phenomenological representation in terms of dynamic
system. This first representation for Na and K channels of giant squid
axon was supposed by Hodjkin&Huxley in 1952.
9. However, the H&H model has not key properties of neuronal activity.
To avoid this disadvantage, this model may be widened by additional
ion channels. Moreover, the cell body may be divided into
compartments.
10.Using the cable model for description of dendrite arbor had blocked
the researches of distal synapse influence for ten years up to 80s and
allows to model cell activity in dependence of its geometry.
11.There are many types of neuronal activity and different classifications.
12.The most of accuracy classification methods use pure mathematical
formalizations.
13.Identification of network environment is complicated experimental
problem that was resolved just recently. The simple example shows
that one connection can dramatically change the pattern of neuron
output.

Phenomenological models of neuron
Is it possible to model only phenomena of neuronal activity
without detailed consideration of electrical genesis?

Hodjkin-Huxley style models
Reduction of base equations or/and number of compartments
or/and simplification of equations for currents
Speed up and dimension of network
Accuracy neuron description
Simplification
Sophistication
Description of neuron dynamics by formal function
Integrate-and-Fire style models

FitzHugh-Nagumo's model
R. FitzHugh
«Impulses and physiological states in models of nerve membrane»
Biophys. J., vol. 1, pp. 445-466, 1961.
2 3
v '=ab vc v d v −u u' =  e v−u 

Izhikevich's model
Eugene M. Izhikevich
«Which Model to Use for Cortical Spiking Neurons?»
IEEE TRANSACTIONS ON NEURAL NETWORKS, VOL. 15, NO. 5, SEPTEMBER 2004
2
v ' =0.04 v 5 v140−u
u ' =ab v−u where a,b,c,d – model parameters
if v30 then v=c ,u=ud

Izhikevich's model

Integrate-and-Fire model
Simple integrator:
du
 = ∑ I syn−ut 
⌠ dt
│dt
⌡
Threshold function – short circuit of membrane:
if u then u=0

Integrate-and-Fire model
Simple integrator:
du
 = ∑ I syn−ut 
⌠ dt
│dt
⌡
Threshold function – short circuit of membrane:
if u then u=0

Modified Integrate-and-Fire model
Master and slave integrators
dut  r du ap 1
 =rI t  uap t −ut  −ut ap =  u t −uap t 
dt r ap dt ap
Adaptive threshold
{
a
dui t  r  ut −ui t  if utui t 
= =ui t cth
dt a
f
 ut−ui t if ut ui t 
Pulse generator:
 du (t ) 1 u ap (t ) − u (t ) u (t ) 2U s τ fire
 = I (t ) + − + если t − t ' <
dt C CRap τs τ fire 2

 du (t ) 1
 u ap (t ) − u (t ) u (t ) 2U s τ fire
 = I (t ) + − − если < t − t ' < τ fire
dt C CRap τs τ fire 2

 du (t ) 1 u ap (t ) − u (t ) u (t )
 = I (t ) + − во всех остальных случаях
 dt
 C CRap τ

Modified Integrate-and-Fire model

Modified Integrate-and-Fire model

omparative characteristics of
neuron models by Izhikevich

Synapses: chemical and electrical

Synapses: chemical and electrical

Chemical synapse models (ion model)
g s t =g s t s −t  Phenomenological models
I s =g s  u−E s 
g s t =g s u ps ,t  g s t =g s u ps ,t ,[ Ma2 + ]o ,
u ps , t= P u ps , t 
ps 1
u , t=1−
1exp 
u ps −
 
ps 1 ps 2+ ps
u , t=1− u , t ,[ Ma ]o =u ,t  g ∞
1exp 
u ps t− t−
   
g ∞= 1exp
 u[ Ma ]o

2+
−1


Chemical synapse models
(Phenomenological models)
{
if tt s
{
0 if tt
s 0
{
s
0 if tt t s −t t s −t
I s = t −t
e
s
if other
I s=

t s −t
 
exp 1−
t s −t
  if other
I s=

e
1
−e
1−2
2
if other
{
m s mi
− if t−t sr
dmi t  r  f
I s = mi t =
dt mi
− if t−t sr
f

Learning, memory and neural networks
Gerald M. Edelman
The Group-Selective
Theory of Higher Brain
Function
The brain is hierarchy of non-degenerate
neural group

Learning, memory and neural networks
Sporns O., Tononi G.,
Edelman G.M.
Theoretical Neuroanatomy:
Relationg Anatomical and
Functional Connectivity in
Graphs and Cortical
Connection Matrices
Cerebral Cortex, Feb 2000;
10: 127 - 141

Learning, memory and neural networks
Gerald M. Edelman – Brain Based Device (BBD)
Krichmar J.L., Edelman G.M.
Machine Psychology: Autonomous Behavior,
Perceptual Categorization and Conditioning in a
Brain-based Device
Cerebral Cortex Aug. 2002; v12: n8 818-830

Learning, memory and neural networks
Gerald M. Edelman – Brain Based Device (BBD)
McKinstry J.L., Edelman G.M.,
Krichmar J.K.
An Embodied Cerebellar Model
for Predictive Motor Control
Using Delayed Eligibility Traces
Computational Neurosci. Conf.
2006

Learning, memory and single neuron
Donald O. Hebb

Learning, memory and single neuron
Guo-qiang Bi and Mu-ming Poo
Synaptic Modifications in
Cultured Hippocampal Neurons:
Dependence on Spike Timing,
Synaptic Strength, and
Postsynaptic Cell Type
The Journal of Neuroscience,
1998, 18(24):10464–1047
Long Term Depression Long-Term Potentiation Spike Time-Dependent Plasticity
(LTD) (LTP) (STDP)

Learning, memory and single neuron
Gerald M. Edelman – Experimental research
Vanderklish P.W., Krushel L.A., Holst B.H., Gally J. A., Crossin K.L., Edelman
G.M.
Marking synaptic activity in dendritic spines with a calpain substrate exhibiting
fluorescence resonance energy transfer
PNAS, February 29, 2000, vol. 97, no. 5, p.2253 2258

Learning and local calcium dynamics
Feldman D.E.
Timing-Based LTP and LTD at
Vertical Inputs
to Layer II/III Pyramidal Cells in
Rat Barrel Cortex
Neuron, Vol. 27, 45–56, (2000)

Learning and local calcium dynamics
Shouval H.Z., Bear
M.F.,Cooper L.N.
A unified model of NMDA
receptor-dependent
bidirectional synaptic plasticity
PNAS August 6, 2002 vol. 99
no. 16 10831–10836

Learning and local calcium dynamics
Mizuno T., KanazawaI., Sakurai M.
Differential induction of LTP and LTD is not
determined
solely by instantaneous calcium concentration: an
essential involvement of a temporal factor
European Journal of Neuroscience, Vol. 14, pp.
701-708, 2001
Kitajima T., Hara K.
A generalized Hebbian rule for activity-
dependent synaptic modification
Neural Network, 13(2000) 445 - 454

Learning and local calcium dynamics

Learning and local calcium dynamics
Urakubo H., Honda M., Froemke R.C., Kuroda S.
Requirement of an Allosteric Kinetics of NMDA Receptors for Spike Timing-Dependent Plasticity
The Journal of Neuroscience, March 26, 2008 v. 28(13):3310 –3323

Learning and local calcium dynamics
Letzkus J.J., Kampa B.M., Stuart
G.J.
Learning Rules for Spike Timing-
Dependent Plasticity
Depend on Dendritic Synapse
Location
The Journal of Neuroscience, 2006
26(41):10420 –1042

Learning and local calcium dynamics
Letzkus J.J., Kampa B.M., Stuart
G.J.
Learning Rules for Spike Timing-
Dependent Plasticity
Depend on Dendritic Synapse
Location
The Journal of Neuroscience, 2006
26(41):10420 –1042

Learning and Memory
Open issues
Frey & Morris, 1997

Learning and Memory
Open issues
from: Frankland & Bontempi (2005)