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AACIMP 2010 Summer School lecture by Ruben Tikidji-Hamburyan. "Physics, Chemistry and Living Systems" stream. "Introduction to Modern Methods and Tools for Biologically Plausible Modeling of Neurons and Neural Networks" course. Part 1.More info at http://summerschool.ssa.org.ua
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Southern Federal University
A.B.Kogan Research Institute for Neurocybernetics
Laboratory for Detailed Analysis and Modeling of Neurons and Neural Networks
Ruben A. Tikidji – [email protected]
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