1 2. Neurons and Conductance-Based Models Lecture Notes on Brain and Computation Byoung-Tak Zhang...

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2. Neurons and Conductance-Based Models

Lecture Notes on Brain and Computation

Byoung-Tak Zhang

Biointelligence Laboratory

School of Computer Science and Engineering

Graduate Programs in Cognitive Science, Brain Science and Bioinformatics

Brain-Mind-Behavior Concentration Program

Seoul National University

E-mail: [email protected]

This material is available online at http://bi.snu.ac.kr/

Fundamentals of Computational Neuroscience, T. P . Trappenberg, 2002.

(C) 2009 SNU CSE Biointelligence Lab(C) 2009 SNU CSE Biointelligence Lab, http://bi.snu.ac.kr 2

Outline2.12.22.32.4

2.5

2.6

Modeling biological neuronsNeurons are specialized cellsBasic synaptic mechanismsThe generation of action potentials: Hodgkin-Huxley equationsDendritic trees, the propagation of action potentials, and compartmental modelsAbove and beyond the Hodgkin-Huxley neuron: fatigue, bursting, and simplifica-tions

(C) 2009 SNU CSE Biointelligence Lab(C) 2009 SNU CSE Biointelligence Lab, http://bi.snu.ac.kr

2.1 Modeling biological neurons

The networks of neuron-like elements¨ The heart of many information-processing abilities of brain

The working of single neurons¨ Information transmission

Simplified versions of the real neurons¨ Make computations with large numbers of neurons tractable¨ Enable certain emergent properties in networks¨ Nodes

The sophisticated computational abilities of neurons The computational approaches used to describe single neu-

rons

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2.2 Neurons are specialized cells

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2.2.1 Structural properties (1)

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Fig. 2.1 (A) Schematic neuron that is similar in appear-ance to pyramidal cells in the neocortex. The compo-nents outlined in the drawing are, however, typical for major neuron types.


2.2.1 Structural properties (2) Fig. 2.1 (B-E) Examples of mor-

phologies of different neurons. ¨ (B) Pyramidal cell from the mo-

tor cortex¨ (C) Granule neuron from olfac-

tory bulb¨ (D) Spiny neuron from the cau-

date nucleus¨ (E) Golgi-stained Purkinje cell

from the cerebellum.

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2.2.2 Information-processing mechanisms

Neurons can receive signals from many other neurons

Synapses (contact site)¨ Presynaptic (from axon)¨ Postsynaptic (to dendrite or cell

body)

Signal = action potential¨ Electronic pulse

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2.2.3 Membrane potential Membrane potential

¨ The difference between the electric potential within the cell and its surround-ing

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mV


2.2.4 Ion channels (1) The permeability of the membrane to certain ions is achieved by ion channels

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Fig. 2.2 Schematic illustrations of dif-ferent types of ion channels. (A) Leak-age channels are always open. (B) Neu-rotransmitter-gated ion channel that opens when a neurotransmitter mole-cule binds to the channel protein, which in turn changes the shape of the channel protein so that specific ions can pass through. (C) The opening of voltage-gated ion channels depends on the membrane potential. This is indicated by a little wire inside the neurons and a grounding wire outside the neuron. Such ion channels can in addition be neurotransmitter-gated (not shown in this figure). (D) Ion pumps are ion channels that transport ions against the concentration gradients. (E) Some neu-rotransmitters bind to receptor mole-cules which triggers a whole cascade of chemical reactions in neurons which produce secondary messengers which in turn can influence ion channels


2.2.4 Ion channels (2) Major ion channels

¨ Pump: use energy¨ Channel: use difference of ions concentration

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2.2.4 Ion channels (3) Resting potential

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mVVrest 65


SupplementEquilibrium potential and Nernst equation

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2.3 Basic synaptic mechanisms Signal transduction within the cell is mediated by electrical

potentials.¨ Electrical synapse or gap-junctions¨ Chemical synapse

Synaptic plasticity

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2.3.1 Chemical synapses and neurotransmitters neurotransmitters stored in

synaptic vesicles¨ glutamate (Glu)¨ gamma-aminobutyric acid

(GABA)¨ Dopamine (DA)¨ acetylcholine (ACh)

synaptic cleft (a small gap of only a few micrometers)

Receptor (channel) and Postsy-naptic potential (PSP)

Fig. 2.3 (A) Schematic illustration of chemical synapses and (B) an electron microscope photo of a synaptic terminal.


2.3.2 Excitatory and inhibitory synapses

Different types of neurotransmitters Excitatory synapse

¨ PSP: depolarization¨ Neurotransmitters trigger the increase of the membrane potential¨ Neurotransmitter: Glu, ACh

Inhibitory synapse¨ PSP: hyperpolarization¨ Neurotransmitters trigger the decrease of the membrane potential¨ Neurotransmitter: GABA

Non-standard synapses¨ Influence ion channels in an indirect way¨ Modulation

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2.3.3 The time-course of postsynaptic potentials

Excitatory postsynaptic potential (EPSP) resulting from non-NMDA re-ceptors

w: amplitude factor¨ strength of EPSP or efficiency of the

synapse f(t)=t·exp(-t): α-function

¨ functional form of a PSP Inhibitory postsynaptic potential

(IPSP) resulting from non-NMDA re-ceptors

EPSP resulting from NMDA receptor

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peakttNMDAnonm etwV /

21 //)( ttm

NMDAm eeVcV

peakttNMDAnonm etwV /

(2.1)

(2.2)

Fig. 2.4 Examples of an EPSP (solid line) and IPSP (dashed line) as modeled by an α-function after the synaptic delay. The dotted lines represent two examples of the difference of two exponential functions with different amplitudes. Note that the time scale is variable and not meant to fit experimen-tal data in the illustration. Indeed, NMDA synapses are often much slower than non-NMDA synapses, often showing their maximal value long after the peak in the effect of the non-NMDA synapses.


2.3.4 Superposition of PSP

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Electrical potentials have the physical property¨ They superimpose as the sum of individual potentials.

Linear superposition of synaptic input Nonlinear voltage-current relationship Nonlinear interaction

¨ Divisive ¨ Shunting inhibition


2.4 The generation of action potentials

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Fig. 2.6 Schematic illustration o the minimal mechanisms necessary for the generation of a spike. The resting poten-tial of a cell is maintained by a leakage channel through which potassium ions can flow as a result of concentration differences between the inside of the cell and the surround-ing fluid. A voltage-gated sodium channel allows the in-flux of positively charged sodium ions and thereby the de-polarization of the cell. After a short time the sodium channel is blocked and a voltage-gated potassium channel opens. This results in a hyperpolarization of the cell. Fi-nally, the hyperpolarization causes the inactivation of the voltage-gated channels and a return to the resting potential.

Fig. 2.5 Typical form of an action potential, redrawn from an oscilloscope picture of Hodgkin and Huxley.


2.4.3 Hodgkin-Huxley equations (1)

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Quantified the process of spike generation

Input Current ¨ Electric conductance¨ Membrane potential relative to

the resting potential ¨ Equilibrium potential

K+, Na+ conductance dependent ¨ n, The activation of the K channel¨ m, The activation of the Na channel¨ h, The inactivation of the Na channel

)( ionionion EVgI

ionE

iong

V

4ngg KK

(2.3)

(2.4)

(2.5)hmgg NaNa3

Fig. 2.7 A Circuit representation of the Hodgkin-Huxley model. This circuit includes a capacitor on which the membrane potential can b e measured and three resistors together with their own battery modeling the ion channels; two are voltage-dependent and one is static.



n, m, h have the same form of first-order differential-equation x should be substituted by each of the variables n, m and h

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)]([)(

10 Vxx

Vdt

dx

x

(2.6)

Fig. 2.8 (A) The equilibrium functions and (B) time constants for the three variables n, m, and h in the Hodgkin-Huxley model with parameters used to model the gain axon of the squid.



Hodgkin-Huxley model¨ C, capacitance¨ I(t), external current

Three ionic currents

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ion

ion tIIdt

dVC )(

(2.7)

)()()()( 34 tIEVgEVhmgEVngdt

dVC LLNaNaKK

)]([ 0 Vnndt

dnn

)]([ 0 Vmmdt

dmm

)]([ 0 Vhhdt

dhh

(2.8)

(2.9)

(2.10)

Fig. 2.7 A Circuit represen-tation of the Hodgkin-Hux-ley model. This circuit in-cludes a capacitor on which the membrane potential can b e measured and three re-sistors together with their own battery modeling the ion channels; two are volt-age-dependent and one is static.


2.4.4 Numerical integration

A. A Hodgkin-Huxley neuron responds with constant firing to a constant external current.

B. The dependence of the firing rate with the external current (nonlinear curve). (dashed line: noise added)

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Fig. 2.9 (A) A Hodgkin-Huxley neuron responds with constant firing to a constant external cur-rent of Iext = 10. (B) The depen-dence of the firing rate with the strength of the external current shows a sharp onset of firing around Ic

ext = 6 (solid line). High-frequency noise in the current has the tendency to ‘linearize’ these response characteristics (dashed line).


2.4.5 Refractory period Absolute refractory period

¨ The inactivation of the sodium channel makes it impossible to initiate another action potential for about 1ms.

¨ Limiting the firing rates of neu-rons to a maximum of about 1000Hz

Relative refractory period¨ Due to the hyperpolarizing action

potential it is relatively difficult to initiate another spike during this time period.

¨ Reduced the firing frequency of neurons even further

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2.4.6 Propagation of action potentials

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2.5 Dendritic trees, the propagation of action po-tentials, and compartmental models

Axons with active membranes able to generate action poten-tial

But, dendirtes are a bit more like passive conductors¨ The long cable¨ Cable theory

Compartmental models

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Fig. 2.10 Short cylindrical compartments describ-ing small equipotential pieces of a passive den-drite or small pieces of a active dendrite or axon when including the necessary ion channels. (A) Chains and (B) branches determine the boundary conditions of each compartment. (C) The topol-ogy of single neurons can be reconstructed with compartments, and such models can be simulated using numerical integration techniques.


2.5.1 Cable theory (1) Cable equation : the spatial-temporal variation of an electric

potential

The potential of the cable at each location of the cable, the physical properties of the cable and has the dimensions of

Ωcm,¨ Ex) cylindrical cable of diameter d,¨ The specific resistance of the membrane,¨ The specific intracellular resistance of the cable,

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),(4),(),(),(

2

22 txIRdVtxV

t

txV

x

txVinjmrestm

mm

i

m

R

dR

4

),( txVm

mR

iR

(2.11)

(2.12)


2.5.1 Cable theory (2) The time constant

¨ The resistance of the membrane,¨ The capacitance, the capacitance per unit area,

Stable configuration

Semi-infinite cable

Nonlinear cable equation, include voltage-dependent ion channels as in the Hodgkin-Huxley equation

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mmm CR

mR

mC

xxm evevxV

21)(

restx

restm VeVVV )( 0

)),,((4),(),(),(

2

22 ttxVIRdVtxV

t

txV

x

txVminjmrestm

mm

(2.13)

(2.14)

(2.15)

(2.16)


2.5.2 Physical shape of neurons The physical shape of neurons

¨ Simple homogeneous linear cable is to divide the cable into small pieces, compartment.

¨ Each compartment is governed by a cable equation for a finite cable

¨ The boundary conditions

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2

11

2

2

)(

)()(2),(

jij

jjjm

xx

tVtVV

x

txV

(2.17)


2.5.3 Neuron simulators Ex) Neuron, Genesis

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Fig. 2.11 Example of the NEURON simulation environment. The example shows a simulation with a re-constructed pyramidal cell, in which a short current pulse was injected at t = 1 ms at the location of the dendrite indicated by the dot and the cursor in the window on the left.


2.6 Above and beyond the Hodgkin-Huxley neuron: fatigue, bursting, simplifications (1)

Simplification of the Hodgkin-Huxley model¨ The time constant is so small for all values of .¨ The rate of inactivation of the Na+ channel is approximately re-

ciprocal to the opening of the K+ channel.¨ Neocortical neurons often show no inactivation of the fast Na+

channel.

¨ The recovery of the membrane potential

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)]([ 0 VRRdt

dRR

m V

R

(2.18)

(2.19)

)())(()( tIEVVgEVRgdt

dVC NaNaKK


2.6 Above and beyond the Hodgkin-Huxley neuron: fatigue, bursting, simplifications (2)

Extensions of the Hodgkin-Huxley model¨ Ca2+ channel, a dynamic gating variable T.¨ Slow hyperpolarizing current , Ca2+-mediated K+ channel, a

dynamic gating variable H.

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)()()()())(( tIEVHgEVTgEVRgEVVgdt

dVC hHTTkkNaNa

24810.33476.08.17)( VVVgNa)]([ 0 VRR

dt

dRR

)]([ 0 VTTdt

dTT

)](3[ VTHdt

dHH

240 210.3037.024.1)( VVVR

240 810116.0205.4)( VVVT

(2.20)

(2.21)

(2.22)

(2.23)

(2.24)

(2.25)

(2.26)


2.6 Above and beyond the Hodgkin-Huxley neuron: fatigue, bursting, simplifications (3) Simulation of the Wilson model

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Fig. 2.12 Simulated spike train of the Wilson model. The upper graph simulates fast spiking neu-rons (FS) typical of inhibitory neurons in the mammalian neo-cortex (τR = 1.5ms, gT = 0.25, gH = 0). The middle graph models regular spiking neurons (RS) with longer spikes (τR = 4.2ms, gT = 0.1). The slow calcium-me-diated potassium channel (gH = 5) is responsible for the slow adaptation in the spike fre-quency. The lower graph demon-strate that even more complex behavior, typical of mammalian neocortical neurons, is incorpo-rated in the Wilson model. The parameters (τR = 4.2ms, gT = 2.25, gH = 9.5) result I a typical bursting behavior, including a typical after-depolarizing poten-tial (ADP).


Fig. 2.12 Simulated spike train of the Wilson model. The up-per graph simulates fast spiking neurons (FS) typical of in-hibitory neurons in the mammalian neocortex (τR = 1.5ms, gT = 0.25, gH = 0). The middle graph models regular spiking neurons (RS) with longer spikes (τR = 4.2ms, gT = 0.1). The slow calcium-mediated potassium channel (gH = 5) is respon-sible for the slow adaptation in the spike frequency. The lower graph demonstrate that even more complex behavior, typical of mammalian neocortical neurons, is incorporated in the Wilson model. The parameters (τR = 4.2ms, gT = 2.25, gH = 9.5) result I a typical bursting behavior, including a typical af-ter-depolarizing potential (ADP).

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Conclusion Neurons utilize a variety of specialized biochemical mecha-

nisms for information processing transmission¨ Membrane potential, ion channel¨ Action potential, neurotransmitter¨ Propagation, refractory period

Conductance-based models¨ Hodgkin-Huxley equation

Compartmental models¨ Cable theory

Neuron simulators¨ Neuron, Genesis

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