Southern Federal University

A.B.Kogan Research Institute for Neurocybernetics

Laboratory for Detailed Analysis and Modeling of Neurons and Neural Networks

Ruben A. Tikidji – Hamburyanrth@nisms.krinc.ru

2010

Introduction to modern methods and tools for biologically plausible

modeling of neurons and neural networks

Lecture I

Brain as an object of research● System level – to research the brain as a

whole ● Structure level:

a) anatomicalb) functional

● Populations, modules and ensembles● Cellular● Subcellular

System level

Reception (sense) functions: vision, hearing, touch, ... Perception.

Cognitive functions: attention, memory, emotions, speech, thinking ...

Methods: EEG, PET, MRT, ...

System level

Mathematical Modeling:Population models based on collective dynamicsOscillating networksFormal neural networks, fuzzy logic

Structure level

Anatomical Functional

Methods of research and modelinguse and combine methods of both system and population levels

Populations, modules and ensembles

Research methods:Focal macroelectrode records from intact brainMarking by selective dyesSpecific morphological methods

Populations, modules and ensembles

Modeling methods:Formal neural networksBiologically plausible models:

Population or/and dynamical modelsModels with single cell accuracy (detailed models)

Cellular and subcellular levels

Research methods:Extra- and intracellular microelectrode recordsDyeing, fluorescence and luminescence microscopySlice and culture of tissueGenetic researchResearch with Patch-Clamp methods from cell as a whole up to

selected ion channel Biochemical methods

Cellular and subcellular levels

Modeling methods:Phenomenological models of single neurons and synapsesModels with segmentation and spatial integration of cell bodyModels of neuronal membrane locusModels of dynamics of biophysical and biochemical processes in

synapsesModels of intracellular components and reactionsQuantum models of single ion channels

Is a brain a set of cells or syncytium?

v v

Single Cell

OR

Syncytium

Muscle Cells Liver Cells Heart Cells

Cellular and subcellular levelsRamon-y-Cajal's paradigm.

SantiagoRamon-y-Cajal

1888 – 1891

CamilloGolgi1885

Cellular and subcellular levelsRamon-y-Cajal's paradigm.

Soma of neuron

Dendrite tree or arbor of neuron:the set of neuron inputs

Axon hillock,The impulse generating zone

Axon, the nerve:output of neuron

Neuron as alive biological cell

Spike generation. Afterpolarization

threshold

Afterpolarization

Potential impulse«Action Potential» or Spike

Synapse

Formal description

Σ=

Formal description

= ⌠│dt⌡

⌠│Σ dt⌡

Formal description

Σ= ⌠│Σ dt⌡

Ions in neuron. Reversal potential

NaClC

1=1.5 mM/L

NaClC

2=1.0 mM/L

U

Na+

Na+

Na+

c= RT lnC1

C 2

e= zF U

e= c

U= RTzF

lnC1

C 2

Na+ and K+ currents

out

in

K+

Na+

Inside (mM) Outside (mM) Voltage(mV)50 437 56397 20 -7740 556 -68

Na+

K+

Cl-

Membrane level organization of neuronSirs A. L. Hodgkin, A. F. Huxley and squid with its own giant axon

Membrane level organization of neuronSirs A. L. Hodgkin, A. F. Huxley and squid with its own giant axon

Current of capacitance

When K+ is blocked. Na+ current.

When Na+ is blocked. K+ current.

Ion currents blockage. Spike generation

Ion currents blockage. Spike generation

Gate currents and method Patch-Clamp

Erwin Neherand

Bert Sakmann

Erwin Neherand

Bert Sakmann

Gate currents and method Patch-Clamp

Molecular level. The last outpost of biologically plausible modeling.

-

+-

E

x

Molecular level. The last outpost of biologically plausible modeling.

Hodjkin-Huxley equationsDynamics of gate variables

C dudt= g Ku− E Kg Nau− E NagLu− E L

g Na= gNa m3 hg K= g K n4

dfdt=1− ffu− f fu

where f – n, m and h respectivelydfdt=− 1f − f ∞

u=fufu; f ∞u=fu

fufu=fuu

First activation and inactivation functions.

α(u) β(u)

n0.1− 0.01ue1− 0.1u− 1

2.5− 0.1ue2.5− 0.1u− 1

m2.5− 0.1ue2.5− 0.1u− 1 4e

− u18

h 0.07 e− u20

1e3− 0.1u1

Hodgkin, A. L. and Huxley, A. F. (1952).

A quantitative description of ion currents and its applications to conduction and excitation in nerve membranes.

J. Physiol. (Lond.), 117:500-544.

Citation from:Gerstner and Kistler «Spiking Neuron Models. Single Neurons, Populations, Plasticity» Cambridge University Press, 2002

Threshold is depended upon speed of potential raising

Threshold adaptation under prolongated polarization.

Non-plausibility of the most biologically plausible model!

Non-plausibility of the most biologically plausible model!

The Zoo of Ion ChannelsGerstner and Kistler «Spiking Neuron Models. Single Neurons, Populations, Plasticity»

Cambridge University Press, 2002

C dudt= I i∑ k

I kt

I kt= g k m pk hqku− E k

dmdt=1− mmu− mmu

dndt=1− nnu− nnu

The Zoo of Ion ChannelsGerstner and Kistler «Spiking Neuron Models. Single Neurons, Populations, Plasticity»

Cambridge University Press, 2002

C dudt= I i∑ k

I kt

I kt= g k m pk hqku− E k

dmdt=1− mmu− mmu

dndt=1− nnu− nnu

Regular Spiking (RS) cell (Na, K, M)

Fast Spiking (FS) cell(Na, K)

Intrinsically Bursting(IB) cell (Na, K, M,CaL)

Slow firing (SF) cell(Na, K, h)

Rebound bursting (LTS) cell (Na, K, M,CaT)

Repetitive Bursting (RB) cell (Na, K, M, CaL)

The Zoo of Ion Channels

C dudt=∑ i

g iu− E i

gmu− Emg Au− u'I

Compartment model of neuron

Compartment model of neuron

Cable equationRL ixdx= ut , xdx− ut , x

ixdx− ix=

= C ∂∂ t ut , x1RT

ut , x− I extt , x

C = c dx, RL = r

L dx, R

T-1 = r

T-1 dx, I

ext(t, x) = i

ext(t, x) dx.

∂2

∂ x 2 ut , x= c r L∂∂ t ut , x

r L

rTut , x− r L iextt , x

rL/rT = λ2 и crL = τ ∂∂ t

ut , x=∂2

∂ x 2 ut , x−2 ut , xiextt , x

Fist modeling fault

John Carew Eccles

Wilfrid Rall

Cell geometry and activityixdx− ix= C ∂∂ t ut , x∑

i[g it , uut , x− E i]− I extt , x

∂2

∂ x2 ut , x= c r L∂∂ t

ut , xr L∑i[g it , uut , x− E i]− r L iextt , x

Ion channels from Mainen Z.F., Sejnowski T.J. Influence of dendritic structureon firing pattern inmodelneocortical neurons // Nature, v. 382: 363-366, 1996.

EL= –70, Ena= +50, EK= –90, Eca= +140(mV)Na: m3h: αm= 0.182(u+30)/[1–exp(–(u+30)/9)] βm= –0.124(u+30)/[1–exp((u+30)/9)]

h∞= 1/[1+exp(v+60)/6.2] αh=0.024(u+45)/[1–exp(–(u+45)/5)]βh= –0.0091(u+70)/[1–exp((u+70)/5)]

Ca: m2h: αm= 0.055(u + 27)/[1–exp(–(u+27)/3.8)] βm=0.94exp(–(u+75)/17)αh= 0.000457exp( –(u+13)/50) βh=0.0065/[1+ exp(–(u+15)/28)]

KV: m: αm= 0.02(u – 25)/[1–exp(–(u–25)/9)] βm=–0.002(u – 25)/[1–exp((u–25)/9)]KM: m: αm= 0.001(u+30)/[1-exp(–(u+30)/9)] βm=0.001 (u+30)/[1-exp((u+30)/9)]KCa: m: αm= 0.01[Ca2+]i βm=0.02; [Ca2+]i (mM)[Ca2+]i d[Ca2+]i /dt = –αICa – ([Ca2+]i – [Ca2+]∞)/τ; α=1e5/2F, [Ca2+]∞=0.1μM, τ=200msRaxial 150Ώcm (6.66 mScm)

Cell geometry and activitySoma DendriteNa 20(pS/μm2)Ca 0.3(pS/μm2)KCa 3(pS/μm2)KM 0.1(pS/μm2)KV 200(pS/μm2)L 0.03(mS/cm2)

Na 20(pS/μm2)Ca 0.3(pS/μm2)KCa 3(pS/μm2)KM 0.1(pS/μm2)L 0.03(mS/cm2)

Cell geometry and activity

Neuron types by Nowak et. al. 2003

Neuron types by Nowak et. al. 2003

Bannister A.P.Inter- and intra-laminar connections of pyramidal cells in the neocortexNeuroscience Research 53 (2005) 95–103

How to identify the neurons and connections.

How to identify the neurons and connections.

D. Schubert, R. Kotter, H.J. Luhmann, J.F. StaigerMorphology, Electrophysiology and Functional Input Connectivity of Pyramidal Neurons Characterizes a Genuine Layer Va in the Primary Somatosensory CortexCerebral Cortex (2006);16:223--236

Neurodynamics and circuit of cortex connections

Somogyi P., Tamas G., Lujan R., Buhl E.H.Salient features of synaptic organisation in the cerebral cortexBrain Research Reviews 26 (1998). 113 – 135

Neurodynamics and circuit of cortex connections

West D.C., Mercer A., Kirchhecker S., Morris O.T., Thomson A.M.

Layer 6 Cortico-thalamic Pyramidal CellsPreferentially Innervate Interneurons andGenerate Facilitating EPSPs

Cerebral Cortex February 2006;16:200--211

Thomson A.M., Lamy C. 2007

Neurodynamics and circuit of cortex connections

Properties of single neuron in network and network with such elements

Autoinhibition as nontrivial exampleDodla R., Rinzel J., Recurrent inhibition can enhance spontaneous neuronal firing // CNS 2005

Autoinhibition as nontrivial exampleDodla R., Rinzel J., Recurrent inhibition can enhance spontaneous neuronal firing // CNS 2005

Lyall Watson

If the brain were so simple we could understand it, we would be so simple we couldn't