Basic Models in Theoretical Neuroscience

Oren Shriki

2010

Integrate and Fire and Conductance Based Neurons

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References about neurons as electrical circuits:

• Koch, C. Biophysics of Computation, Oxford Univ. Press, 1998.

• Tuckwell, HC. Introduction to Theoretical Neurobiology, I&II, Cambridge UP, 1988.

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The Neuron as an Electric Circuit

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Intracellular Recording

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Generation of Electric Potential on Nerve Cell Membranes

Chief factors that determine the resting membrane potential:

• The relative permeability of the membrane to different ions

• Differences in ionic concentrations

Ion pumps – Maintain the concentration gradient by actively moving ions against the gradient using metabolic resources.

Ion channels – “Holes” that allow the passage of ions in the direction of the concentration gradient. Some channels are selective for specific ions and some are not selective.

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Ion Channels and Ion Pumps

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The Neuron as an Electric Circuit

• Differences in ionic concentrations Battery

• Cell membrane Capacitor

• Ionic channels Resistors

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The Neuron as an Electric Circuit

Extracellular

Intracellular9

RC circuits

• R – Resistance (in Ohms)

• C – Capacitance (in Farads)

I RCCurrent

source

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RC circuits

I RC

)(tIR

V

dt

dVC

• The dynamical equation is:

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RC circuits• Defining:• We obtain:

• The general solution is:

RC

RtIVdt

dV )(

tIetdeVtVttt

Rt

0

0

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RC circuit• Response to a step current:

0 0

0

0

0 0

( | 1 1

0 1

t tt tt t

t t t t tt

t t

V

I tI t

t

dt e I t e I dt e

e I e e I e I e

V t V e IR e

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RC circuit• Response to a step current:

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The Integrate-and-Fire Neuron

• R – Membrane Resistance (1/conductance)

• C – Membrane Capacitance (in Farads)

I RC

inside

outside

EL

Threshold mechanism

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Integrate-and-Fire Neuron

• If we define:

• The dynamical equation will be:

• To simplify, we define:

• Thus:

outin VVV

)(1

tIEVRdt

dVC L

LEVVV outin

)(tIR

V

dt

dVC

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Integrate-and-Fire Neuron

• The threshold mechanism:

– For V<θ the cell obeys its passive dynamics– For V=θ the cell fires a spike and the voltage resets to

0.

• After voltage reset there is a refractory period, τR.

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Integrate-and-Fire Neuron• Response to a step current:

IR<θ:

t

V

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Integrate-and-Fire Neuron

• Response to a step current: IR>θ:

V

t

T

τR τR τR

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Integrate-and-Fire Neuron

• Finding the firing rate as a function of the applied current:

1

1 1

1 1

1

R

R R

T

T T

RR

R

V t IR e

e eIR IR

Tln T ln

IR IR

T lnIR

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Integrate-and-Fire Neuron 1 1 1

1 1

1

1

R R

LR

L

f IT

ln lnIR IR

gCln

g I

f

I

1

R

cIR

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The Hodgkin-Huxley Equations

n

h

m

/τ(V)-nndn/dt

/τ(V)-hhdh/dt

/τ(V)-mmdm/dt

)(, tIwVIdt

dVC ion

)()()(

,,,

LLK4

KNa3

Na VVgVVngVVhmg

nhmVI ion

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The Hodgkin & Huxley Framework

)(,,, 1 tIWWVIdt

dVC Nion

V

WVW

dt

dW

i

iii

,

Each gating variable obeys the following dynamics:

i

- Represents the effect of temperature

- Time constant

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The Hodgkin & Huxley Framework

)(,,, 1 jNjjj VVWWVgI

The current through each channel has the form:

j

jg - Maximal conductance (when all channels are open)

- Fraction of open channels (can depend on several W variables). 24

The Temperature Parameter Φ

• Allows for taking into account different temperatures.

• Increasing the temperature accelerates the kinetics of the underlying processes.

• However, increasing the temperature does not necessarily increase the excitability. Both increasing and decreasing the temperature can cause the neuron to stop firing.

• A phenomenological model for Φ is:

10/3.6Temp3 25

Hodgkin & Huxley Model

n

h

m

/τ(V)-nndn/dt

/τ(V)-hhdh/dt

/τ(V)-mmdm/dt

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Ionic Conductances During an Action Potential

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Repetitive Firing in Hodgkin–Huxley Model

A: Voltage time courses in response to a step of constant depolarizing current. from bottom to top: Iapp= 5, 15, 50, 100, 200 in μamp/cm2). Scale bar is 10 msec. B: f-I curves for temperatures of 6.3,18.5, 26◦C, as marked. Dotted curves show frequency of the unstable periodic orbits. 28

Fast-Slow Dissection of the Action Potential

• n and h are slow compared to m and V.• Based on this observation, the system can be

dissected into two time-scales.• This simplifies the analysis.• For details see:

Borisyuk A & Rinzel J. Understanding neuronal dynamics by geometrical dissection of minimal models. In, Chow et al, eds: Models and Methods in Neurophysics (Les Houches Summer School 2003), Elsevier, 2005: 19-72.

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Correlation between n and h

• During the action potential the variables n and h vary together.

• Using this correlation one can construct a reduced model.

• The first to observe this was Fitzhugh.

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Simplified Versions of the HH Model

• Models that generate action potentials can be constructed with fewer dynamic variables.

• These models are more amenable for analysis and are useful for learning the basic principles of neuronal excitability.

• We will focus on the model developed by Morris and Lecar.

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The Morris-Lecar Model (1981)

• Developed for studying the barnacle muscle.

• Model equations:

)(, tIwVIdt

dVC appion

V

wVw

dt

dw

w

)()())((, LLKKCaCa VVgVVwgVVVmgwVI ion 32

Morris-Lecar Model• The model contains K and Ca currents.• The variable w represents the fraction of open K

channels.• The Ca conductance is assumed to behave in an

instantaneous manner.

21 /tanh15.0)( VVVVm

43 2/cosh/1 VVVVw

43 /tanh15.0)( VVVVw

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Morris-Lecar Model

• A set of parameters for example:

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2

18

2.1

4

3

2

1

V

V

V

V

2

8

4

L

K

Ca

g

g

g

60

84

120

L

K

Ca

V

V

V

04.0cm

μF20

2

C

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Morris-Lecar Model

• Voltage dependence of the various parameters (at long times):

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Conductance-Based Models of Cortical Neurons

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Conductance-Based Models of Cortical Neurons

• Cortical neurons behave differently than the squid axon that Hodgkin and Huxley investigated.

• Over the years, people developed several variations of the HH model that are more appropriate for describing cortical neurons.

• We will now see an example of a simple model which will later be useful in network simulations.

• The model was developed by playing with the parameters such that its f-I curve is similar to that of cortical neurons.

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Frequency-Current Responses of Cortical Neurons

Excitatory Neuron:

Ahmed et. al., Cerebral Cortex 8, 462-476, 1998

Inhibitory Neurons:

Azouz et. al., Cerebral Cortex 7, 534-545, 1997

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Frequency-Current Responses of Cortical Neurons

The experimental findings show what f-I curves of cortical neurons are:

• Continuous – starting from zero frequency.• Semi-Linear – above the threshold current the

curve is linear on a wide range.

How can we reconstruct this behavior in a model?

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An HH Neuron with a Linear f-I Curve

Shriki et al., Neural Computation 15, 1809–1841 (2003) 40

Linearization of the f-I Curve

• We start with an HH neuron that has a continuous f-I curve (type I, saddle-node bifurcation).

• The linearization is made possible by the addition of a certain K-current called A-current.

• The curve becomes linear only when the time constant of the A-current is slow enough (~20 msec).

• There are other mechanisms for linearizing f-I curves.

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Model Equations:

Shriki et al., Neural Computation 15, 1809–1841 (2003)

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