single neuron models (1)

Single Neuron Models (1)

LECTURE 3

I. Overview

II. Single-Compartment Models − Integrate-and-Fire Models − Firing rate models

− The Hodgkin-Huxley Model − Synaptic conductance description

− The Runge-Kutta method

III. Multi-Compartment Models − Two-Compartment Models

Detailed descriptions involving thousands of coupled differential equations are useful

for channel-level investigation

Greatly simplified caricatures are useful for analysis and studying large

interconnected networks

From compartmental models to point neurons

Axon hillock

I. Overview

II. Single-Compartment Models − Integrate-and-Fire Models − Firing rate models

− The Hodgkin-Huxley Model − Synaptic conductance description

− The Runge-Kutta method

III. Multi-Compartment Models − Two-Compartment Models

The equivalent circuit for a genericone-compartment model

A

Ii

dt

dVc e

mm

A

Ii

dt

dVc

QVc

emm

m

H-H model

Passive or leaky integrate-and-fire model(…/cm2)

I. Overview

II. Single-Compartment Models − Integrate-and-Fire Models − Firing rate models

− The Hodgkin-Huxley Model − Synaptic conductance description

− The Runge-Kutta method

III. Multi-Compartment Models − Two-Compartment Models

• Maybe the most popular neural model

• One of the oldest models (Lapicque 1907)

(Action potentials are generated when the integrated sensory or synaptic inputs to a neuron reach a threshold value)

• Although very simple, captures almost all of the important properties of the cortical neuron

• Divides the dynamics of the neuron into two regimes– Sub- Threshold– Supra- Threshold

• Sub Threshold:

－ Linear ODE － Without input ( ), the stable fixed point

at ( )LEV

0eI

emLm IRVEdt

dV

A

IEVg

dt

dVc e

LLm )(

(τm = RmCm = rmcm)

• Supra- Threshold:– The shape of the action potentials are more or less

the same– At the synapse, the action potential events translate

into transmitter release– As far as neuronal communication is concerned, the

exact shape of the action potentials is not important,

rather its time of occurrence is important

• Supra- Threshold:– If the voltage hits the threshold at time t0:

• a spike at time t0 will be registered• The membrane potential will be reset to a reset

value (Vreset)• The system will remain there for a refractory period

(t ref)

t0

Vth

Vreset

V

t

resetref

kk

th

emLm

VtttV

tttVV(t)

IRVEdt

dVth : V(t)

]) ,([

)(spikes registered if

if

emLm IRVEdt

dV

Formula summary

I. Overview

II. Single-Compartment Models − Integrate-and-Fire Models − Firing rate models

− The Hodgkin-Huxley Model − Synaptic conductance description

− The Runge-Kutta method

III. Multi-Compartment Models − Two-Compartment Models

Under the assumption:

The information is coded by the firing rate of the neurons and individual spikes are not important

We have:

resetref

kk

th

emLm

VtttV

tttVV(t)

IRVEdt

dVth : V(t)

]) ,([

)(spikes registered if

if

emLm IRVEdt

dV

• The firing rate is a function of the membrane voltage

• g is usually a monotonically increasing function. These models mostly differ in the choice of g.

f g

Sigmoid function

if 0,

if 0)(

th

th

VVaaV

VVVg

V

f

I

f

100 HzPhysiological

Range

• Linear-Threshold model:

)( , VgfIRVEdt

dVemLm

• Based on the observation of the gain function in cortical neurons:

I. Overview

II. Single-Compartment Models − Integrate-and-Fire Models − Firing rate models

− The Hodgkin-Huxley Model − Synaptic conductance description

− The Runge-Kutta method

III. Multi-Compartment Models − Two-Compartment Models

Nobel Prize in Physiology or Medicine in 1963

• Combination of experiments, theoretical hypotheses, data fitting and model prediction

• Empirical model to describe generation of action potentials

• Published in the Journal of Physiology in 1952 in a series of 5 articles (with Bernard Katz)

Stochastic channel

A single ion channel (synaptic receptor channel) sensitive to the neurotransmitter acetylcholine at a holding potential of -140 mV.

(From Hille, 1992)

Single-channel probabilistic formulations

Macroscopic deterministic descriptions

(μS/mm2 mS/mm2)

)( ii EVgi

iii Pgg

the conductance of an open channel × the density of channels in the membrane × the fraction of channels that are open at that time

Persistent or noninactivating conductances

PK = nk

a gating or an activation variable

Activation of the conductance: Opening of the gate

Deactivation: gate closing

(k = 4)

Channel kinetics

nVnVdt

dnnn )()1)((

)()(

1)(

VVV

nnn

nVndt

dnVn )()(

)()(

)()(

VV

VVn

nn

n

opening rate

closing rate

For a fixed voltage V, n approaches the limiting value n∞(V) exponentially with time constant τn(V)

open closed n (1-n))(Vn

)(Vn

For the delayed-rectifier K+ conductance

Transient conductances

PNa = mkh

activation variable

(k = 3)

inactivation variable

zVzVdt

dzzz )()1)((

m or h

The Hodgkin-Huxley Model

A

Ii

dt

dVc e

mm

zVzdt

dzVz )()( Gating equation

The voltage-dependent functions of the Hodgkin-Huxley model

deinactivation

inactivation

activation

deactivation

Improving Hodgkin-Huxley ModelImproving Hodgkin-Huxley Model

Connor-Stevens Model (HH + transient

A-current K+) (EA~ EK)

transient Ca2+ conductance

(L, T, N, and P types.ECaT = 120mV)

Ca2+-dependent K+ conductance

－ spike-rate adaptation

－ type I behavior (continuous firing rate)

－ Ca2+ spike, burst spiking, thalamic relay neurons

I. Overview

II. Single-Compartment Models − Integrate-and-Fire Models − Firing rate models

− The Hodgkin-Huxley Model − Synaptic conductance description

− The Runge-Kutta method

III. Multi-Compartment Models − Two-Compartment Models

Synaptic conductances

Synaptic open probability

Transmitter release probability

)( sss EVgi

Two broad classes of synaptic conductances

Metabotropic: Many neuromodulators including serotonin, dopamine, norepinephrine, and acetylcholine. GABAB receptors.

Ionotropic: AMPA, NMDA, and GABAA receptors

γ-aminobutyric acid

Glutamate, Es = 0mV

Inhibitory and excitatory synapses

Inhibitory synapses: reversal potentials being less than the threshold for action potential generation (GABAA , Es = -80mV)

Excitatory synapses: those with more depolarizing reversal potentials (AMPA, NMDA, Es = 0mV)

The postsynaptic conductance

T = 1ms

A fit of the model to the average EPSC recorded from mossy fiber input to a CA3 pyramidal cell in a hippocampal slice preparation

(Dayan and Abbott 2001)

NMDA receptor conductance

1. When the postsynaptic neuron is near its resting potential, NMDA receptors are blocked by Mg2+ ions. To activate the conductance, the postsynaptic neuron must be depolarized to knock out the blocking ions

2. The opening of NMDA receptor channels requires both pre- and postsynaptic depolarization (synaptic modification)

(Dayan and Abbott 2001)

Synapses On Integrate-and-Fire Neurons

emLm IRVEdt

dV

I. Overview

II. Single-Compartment Models − Integrate-and-Fire Models − Firing rate models

− The Hodgkin-Huxley Model − Synaptic conductance description

− The Runge-Kutta method

III. Multi-Compartment Models − Two-Compartment Models

The Runge-Kutta method (simple and robust)

Then, the RK4 method is given as follows:

An initial value problem:

where yn + 1 is the RK4 approximation of y(tn + 1), and

Program in Matlab or C

作业及思考题

1. 已知参数 EL = Vreset =−65 mV, Vth =−50 mV, τm = 10 ms, and Rm = 10 MΩ ，在 step 电流及其他不同电流注射下，计算模拟整合－发放神经元模型。

2. 写出 Hodgkin-Huxley Model 方程，说明各参数生物学意义。

3. NMDA 受体电导有哪些特性 ?