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Single Neuron Models (1) LECTURE 3

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Page 1: Single Neuron Models (1) LECTURE 3. I.Overview II.Single-Compartment Models − Integrate-and-Fire Models − Firing rate models − The Hodgkin-Huxley Model

Single Neuron Models (1)

LECTURE 3

Page 2: Single Neuron Models (1) LECTURE 3. I.Overview II.Single-Compartment Models − Integrate-and-Fire Models − Firing rate models − The Hodgkin-Huxley Model

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

Page 3: Single Neuron Models (1) LECTURE 3. I.Overview II.Single-Compartment Models − Integrate-and-Fire Models − Firing rate models − The Hodgkin-Huxley Model

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

Page 4: Single Neuron Models (1) LECTURE 3. I.Overview II.Single-Compartment Models − Integrate-and-Fire Models − Firing rate models − The Hodgkin-Huxley Model

From compartmental models to point neurons

Axon hillock

Page 5: Single Neuron Models (1) LECTURE 3. I.Overview II.Single-Compartment Models − Integrate-and-Fire Models − Firing rate models − The Hodgkin-Huxley Model

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

Page 6: Single Neuron Models (1) LECTURE 3. I.Overview II.Single-Compartment Models − Integrate-and-Fire Models − Firing rate models − The Hodgkin-Huxley Model

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)

Page 7: Single Neuron Models (1) LECTURE 3. I.Overview II.Single-Compartment Models − Integrate-and-Fire Models − Firing rate models − The Hodgkin-Huxley Model

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

Page 8: Single Neuron Models (1) LECTURE 3. I.Overview II.Single-Compartment Models − Integrate-and-Fire Models − Firing rate models − The Hodgkin-Huxley Model

• 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

Page 9: Single Neuron Models (1) LECTURE 3. I.Overview II.Single-Compartment Models − Integrate-and-Fire Models − Firing rate models − The Hodgkin-Huxley Model

• 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)

Page 10: Single Neuron Models (1) LECTURE 3. I.Overview II.Single-Compartment Models − Integrate-and-Fire Models − Firing rate models − The Hodgkin-Huxley Model

• 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

Page 11: Single Neuron Models (1) LECTURE 3. I.Overview II.Single-Compartment Models − Integrate-and-Fire Models − Firing rate models − The Hodgkin-Huxley Model

• 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

Page 12: Single Neuron Models (1) LECTURE 3. I.Overview II.Single-Compartment Models − Integrate-and-Fire Models − Firing rate models − The Hodgkin-Huxley Model

resetref

kk

th

emLm

VtttV

tttVV(t)

IRVEdt

dVth : V(t)

]) ,([

)(spikes registered if

if

emLm IRVEdt

dV

Formula summary

Page 13: Single Neuron Models (1) LECTURE 3. I.Overview II.Single-Compartment Models − Integrate-and-Fire Models − Firing rate models − The Hodgkin-Huxley Model

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

Page 14: Single Neuron Models (1) LECTURE 3. I.Overview II.Single-Compartment Models − Integrate-and-Fire Models − Firing rate models − The Hodgkin-Huxley Model

Under the assumption:

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

We have:

Page 15: Single Neuron Models (1) LECTURE 3. I.Overview II.Single-Compartment Models − Integrate-and-Fire Models − Firing rate models − The Hodgkin-Huxley Model

resetref

kk

th

emLm

VtttV

tttVV(t)

IRVEdt

dVth : V(t)

]) ,([

)(spikes registered if

if

emLm IRVEdt

dV

Page 16: Single Neuron Models (1) LECTURE 3. I.Overview II.Single-Compartment Models − Integrate-and-Fire Models − Firing rate models − The Hodgkin-Huxley Model

• 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

Page 17: Single Neuron Models (1) LECTURE 3. I.Overview II.Single-Compartment Models − Integrate-and-Fire Models − Firing rate models − The Hodgkin-Huxley Model

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:

Page 18: Single Neuron Models (1) LECTURE 3. I.Overview II.Single-Compartment Models − Integrate-and-Fire Models − Firing rate models − The Hodgkin-Huxley Model

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

Page 19: Single Neuron Models (1) LECTURE 3. I.Overview II.Single-Compartment Models − Integrate-and-Fire Models − Firing rate models − The Hodgkin-Huxley Model

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)

Page 20: Single Neuron Models (1) LECTURE 3. I.Overview II.Single-Compartment Models − Integrate-and-Fire Models − Firing rate models − The Hodgkin-Huxley Model

Stochastic channel

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

(From Hille, 1992)

Page 21: Single Neuron Models (1) LECTURE 3. I.Overview II.Single-Compartment Models − Integrate-and-Fire Models − Firing rate models − The Hodgkin-Huxley Model

Single-channel probabilistic formulations

Macroscopic deterministic descriptions

Page 22: Single Neuron Models (1) LECTURE 3. I.Overview II.Single-Compartment Models − Integrate-and-Fire Models − Firing rate models − The Hodgkin-Huxley Model

(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

Page 23: Single Neuron Models (1) LECTURE 3. I.Overview II.Single-Compartment Models − Integrate-and-Fire Models − Firing rate models − The Hodgkin-Huxley Model

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)

Page 24: Single Neuron Models (1) LECTURE 3. I.Overview II.Single-Compartment Models − Integrate-and-Fire Models − Firing rate models − The Hodgkin-Huxley Model

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)

Page 25: Single Neuron Models (1) LECTURE 3. I.Overview II.Single-Compartment Models − Integrate-and-Fire Models − Firing rate models − The Hodgkin-Huxley Model

open closed n (1-n))(Vn

)(Vn

For the delayed-rectifier K+ conductance

Page 26: Single Neuron Models (1) LECTURE 3. I.Overview II.Single-Compartment Models − Integrate-and-Fire Models − Firing rate models − The Hodgkin-Huxley Model

Transient conductances

PNa = mkh

activation variable

(k = 3)

inactivation variable

Page 27: Single Neuron Models (1) LECTURE 3. I.Overview II.Single-Compartment Models − Integrate-and-Fire Models − Firing rate models − The Hodgkin-Huxley Model

zVzVdt

dzzz )()1)((

m or h

Page 28: Single Neuron Models (1) LECTURE 3. I.Overview II.Single-Compartment Models − Integrate-and-Fire Models − Firing rate models − The Hodgkin-Huxley Model

The Hodgkin-Huxley Model

A

Ii

dt

dVc e

mm

zVzdt

dzVz )()( Gating equation

Page 29: Single Neuron Models (1) LECTURE 3. I.Overview II.Single-Compartment Models − Integrate-and-Fire Models − Firing rate models − The Hodgkin-Huxley Model

The voltage-dependent functions of the Hodgkin-Huxley model

deinactivation

inactivation

activation

deactivation

Page 30: Single Neuron Models (1) LECTURE 3. I.Overview II.Single-Compartment Models − Integrate-and-Fire Models − Firing rate models − The Hodgkin-Huxley Model

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

Page 31: Single Neuron Models (1) LECTURE 3. I.Overview II.Single-Compartment Models − Integrate-and-Fire Models − Firing rate models − The Hodgkin-Huxley Model

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

Page 32: Single Neuron Models (1) LECTURE 3. I.Overview II.Single-Compartment Models − Integrate-and-Fire Models − Firing rate models − The Hodgkin-Huxley Model

Synaptic conductances

Synaptic open probability

Transmitter release probability

)( sss EVgi

Page 33: Single Neuron Models (1) LECTURE 3. I.Overview II.Single-Compartment Models − Integrate-and-Fire Models − Firing rate models − The Hodgkin-Huxley Model

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

Page 34: Single Neuron Models (1) LECTURE 3. I.Overview II.Single-Compartment Models − Integrate-and-Fire Models − Firing rate models − The Hodgkin-Huxley Model

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)

Page 35: Single Neuron Models (1) LECTURE 3. I.Overview II.Single-Compartment Models − Integrate-and-Fire Models − Firing rate models − The Hodgkin-Huxley Model

The postsynaptic conductance

T = 1ms

Page 36: Single Neuron Models (1) LECTURE 3. I.Overview II.Single-Compartment Models − Integrate-and-Fire Models − Firing rate models − The Hodgkin-Huxley Model

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)

Page 37: Single Neuron Models (1) LECTURE 3. I.Overview II.Single-Compartment Models − Integrate-and-Fire Models − Firing rate models − The Hodgkin-Huxley Model

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)

Page 38: Single Neuron Models (1) LECTURE 3. I.Overview II.Single-Compartment Models − Integrate-and-Fire Models − Firing rate models − The Hodgkin-Huxley Model

(Dayan and Abbott 2001)

Page 39: Single Neuron Models (1) LECTURE 3. I.Overview II.Single-Compartment Models − Integrate-and-Fire Models − Firing rate models − The Hodgkin-Huxley Model

Synapses On Integrate-and-Fire Neurons

emLm IRVEdt

dV

Page 40: Single Neuron Models (1) LECTURE 3. I.Overview II.Single-Compartment Models − Integrate-and-Fire Models − Firing rate models − The Hodgkin-Huxley Model

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

Page 41: Single Neuron Models (1) LECTURE 3. I.Overview II.Single-Compartment Models − Integrate-and-Fire Models − Firing rate models − The Hodgkin-Huxley Model

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

Page 42: Single Neuron Models (1) LECTURE 3. I.Overview II.Single-Compartment Models − Integrate-and-Fire Models − Firing rate models − The Hodgkin-Huxley Model

Program in Matlab or C

Page 43: Single Neuron Models (1) LECTURE 3. I.Overview II.Single-Compartment Models − Integrate-and-Fire Models − Firing rate models − The Hodgkin-Huxley Model

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 ?


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