ZUBIN BHUYAN (CSI 11014)NAYANTARA KOTOKY(CSI 11025)NIRUPAM CHOUDHURY(CSI 11033)

Basic Neuro-Engineering

Integrate and Fire Neuron Modeling

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Outline Introduction(neurons and models) Integrate and fire based neuron model Leaky integrate and fire based neuron model Spike-Response Model

Mathematical Formulation Simulating Refractoriness Fitting to Experimental Data Variations of SRM Effects not captured by SRM

Adaptive Exponential Integrate-and-Fire Model Definition Adaptation, Delayed spiking, Voltage Response, Initial

bursting Fitting to real Neurons’ data

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Review of the neuron

Action potential- very rapid change in membrane potential when a nerve cell membrane is stimulated.

Resting potential (typically -70 mV) to some positive value (typically about +30 mV).

Threshold stimulus & threshold potential(generally 5 - 15 mV less negative than the resting potential)

Neuron model

Biological neuron model- mathematical description of the properties of nerve cells.

Artificial neuron model- aims for computational effectiveness.

Artificial neuron abstraction

Consists of- an input with

some synaptic weight vector

an activation function or transfer function inside the neuron determining output.

Oj=f( ∑wijei )

Biological abstraction

In the case of modelling a biological neuron- Physical analogues are used in place of abstractions

such as “weight” and “transfer function’’. Ion current through the cell membrane is described

by a physical time-dependent current I(t) Insulating cell membrane determines a capacitance

Cm.

A neuron responds to such a signal with a change in voltage, or an electrical potential energy difference between the cell and its surroundings, sometimes resulting in a voltage spike called an action potential.

L. F. Abbott*, 21 May 1999, Lapicque’s introduction of the integrate-and-fire model neuron (1907)

Integrate-and-Fire based Neuron Model

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IAF model

One of the earliest models of a neuron. First investigated in 1907 by Louis

Lapicque. Lapicque modeled the neuron using an

electric circuit consisting of a parallel capacitor and resistor.

When the membrane capacitor was charged to certain threshold potential an action potential would be generated the capacitor would discharge

Theoretical idea

In a biologically realistic neural network, it often takes multiple input signals in order for a neuron to propagate a signal.

Multiple input signals goes from one neuron to the next, increasing the effect of one firing by however many connection there are(done by adjusting the weights between each neuron).

Every neuron has a certain threshold at which it goes from stable to firing.

When a cell reaches its threshold and fires, its signal is passed onto the next neuron, which may or may not cause it to fire.

contd…

If the neuron does not fire, its potential will be raised so that if it receives another input signals within a certain time frame, it will be more likely to fire.

If the neuron does fire, then the signal will be propagated onto the next neuron.

After this, the just-fired neuron goes into a refractory state, in which it doesn't respond to or propagate input signals from other neurons.

This increased potential to fire starts to dampen soon after the input is received.

Mathematical representation

A neuron is represented in time by

When an input current is applied, the membrane voltage increases with time until it reaches a constant threshold Vth , at which point a delta function spike occurs and the voltage is reset to its resting potential, after which the model continues to run.

The firing frequency of the model increases linearly without bound as input current increases.

contd…

By introducing a refractory period tref , we limit the firing frequency of a neuron by preventing it from firing during that period.

Firing frequency as a function of a constant input current is:

Shortcoming: It implements no time-dependent memory. If

the model receives a below-threshold signal at some time, it will retain that voltage boost forever until it fires again.

http://lcn.epfl.ch/~gerstner/SPNM/node26.html

Leaky integrate and fire model

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Description

In the leaky integrate-and-fire model, the memory problem is solved by adding a "leak" term to the membrane potential.

It reflects the diffusion of ions that occurs through the membrane when some equilibrium is not reached in the cell.

(1) Rm is the membrane resistance threshold Ith = Vth / Rm

contd…

When input current exceeds threshold Ith , it causes the cell to fire, else it will simply leak out any change in potential.

firing frequency is:

contd…

We multiply equation (1) by R(resistance) and considering Ω=R Cm of the “leaky integrator” to get-

•Izhikevich, E.M. (2001), Resonate-and-fire neurons, Neural Networks, 14:883-894•Izhikevich E.M. (2003), Simple model of spiking neurons, IEEE Transactions On Neural Networks, 14:1569-1572

Spike-Response Model17

Spike-response model..

Generalization of the leaky integrate-and-fire model

Gives a simple description of action potential generation in neurons

Spike response model includes refractoriness

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SRM: Mathematical Formulation The membrane potential in the spike response

model is given by

Here t’ is the firing time of the last spike η describes the form of the action potential Κ the linear response to an input pulse I(t) is a stimulating current

The next spike occurs if the membrane potential u hits a threshold Ɵ(t-t’) from below in which case t’ is updated

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SRM: Mathematical Formulation (..contd)

The threshold Ɵ is not fixed but depends on the time since the last spike threshold is higher immediately after a

spike then it decays back to its resting value

The spike shape η is a function of the time since the last spike It can describe a depolarizing,

hyperpolarizing, or resonating spike-after potential

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SRM: Mathematical Formulation (..contd)

The responsiveness Κ to an input pulse depends on the time since the last spike since many ion channels are open typically the effective membrane time constant

after a spike is shorter The time course of the response Κ can include

a single exponential combinations of exponentials with different

time constants or resonating behavior in form of a delayed

oscillation (This is the case if the standard Hodgkin-Huxley model is

approximated by the Spike Response Model)

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SRM: Simulating Refractoriness Refractoriness can be modelled as a

combination of increased threshold hyperpolarizing afterpotential and reduced responsiveness after a spike

[as observed in real neurons (Badel et al., 2008)]

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SRM: η (..contd)

Example of a spike shape η with rapid reset, followed by a hyperpolarizing action potential, extracted from data

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SRM: η (..contd)

Example of a spike shape η with depolarizing afterpotential, extracted from data

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SRM: Fits to experimental data SRM can be fitted to experimental data

where a neuron is stimulated

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SRM: Fits to experimental data (…contd.)

SRM fits experimental data to a high degree of accuracy

Predicts a large fraction of spikes with a precision of +/-2ms, (Jolivet et al., 2006).

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Variations of SRM: SRM0

A simplified version of the SRM. Does not include a dependence of the

response kernel K upon the time since the last spike

The threshold can be dynamic as before Easier to fit to experimental data than the

full SRM since it needs less data (Jolivet et al. 2006)

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Cumulative Spike Response Model

Refractoriness and adaptation are modeled by the combined effects of the spike afterpotentials of several previous spikes And not the most recent spike

The advantage of the cumulative model is that it accounts for adaptation and bursting

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SRM with a cumulative dynamic threshold

Value of the threshold depends on all previous spikes and not only the most recent one

The threshold Ɵ is calculated as

tk denotes previous moments of spike firing

Ɵo is the value of the threshold at rest v(t-tk) describes the effect of a spike at

time tk

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Noise in the SRM

Noise can be included into the SRM by replacing the strict threshold criterion,

u(t) = Ɵ, by a stochastic process The probability P of firing a spike within a

very short time Δt is P = ρ(t) Δt

where the instantaneous firing rate ρ(t) is a function of the momentary difference between the membrane potential u(t) and the threshold Ɵ(t),

ρ = f(u - Ɵ)

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Effects not captured by a SRM Pharmacological blocking of ion channels

Biophysics of the neuronal membrane is not described explicitly

combined effects of several ion channel are captured

model cannot make predictions about blocking of individual ion channels

Delayed spike initiation due to different amplitude of the input pulse because of the strict threshold criterion

Dependence of the threshold upon the input

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Brette R. and Gerstner W. (2005), Adaptive Exponential Integrate-and-Fire Model as an Effective Description of Neuronal Activity, J. Neurophysiol. 94: 3637 - 3642.

Adaptive Exponential Integrate-and-Fire Model

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AdEx

A spiking neuron model with two equations The first equation describes

the dynamics of the membrane potential includes an activation term with an

exponential voltage dependence Voltage is coupled to a second equation

which describes adaptation Both variables are reset if an action

potential has been triggered

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AdEx: Mathematical Definition

(…contd.)

The model is described by two differential equations

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V is the membrane potentialI the input currentgL the leak conductance

C the membrane capacitanceEL the leak reversal potentialΔT the slope factor

VT the threshold w the adaptation variable

a is the adaptation coupling parameter

τw is the adaptation time constant

AdEx (…contd.)

Exponential nonlinearity describes the process of spike generation and the upswing of the action potential a spike is said to occur at the time tf when the

membrane potential V diverges towards infinity The downswing of the action potential is a

reset of the voltage to a fixed Vr

at t=tf reset V→Vr Also, change the adaption value by b: w = w +b

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AdEx: Adaptation

Adaptation and regular firing of the AdEx model in response to a current step; voltage (top) and adaptation variable (bottom)

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[Naud, Marcille, Colpath, Gerstner (2008), Firing patterns in the adaptive exponential integrate-and-fire model]

AdEx: Voltage Response

Voltage response of the AdEx model to a series of regularly spaced (10 Hz) current pulse

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[Naud, Marcille, Colpath, Gerstner (2008), Firing patterns in the adaptive exponential integrate-and-fire model]

Voltage (X-axis) and adaptation variable

Resting potential marked by cross reset values marked by squares

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AdEx: Initial bursting as response

Voltage as a function of time

[Naud, Marcille, Colpath, Gerstner (2008), Firing patterns in the adaptive exponential integrate-and-fire model]

Bursting with 3 spikes per burst in the AdEx model Bursting occurs when the reset value Vr is high, so that spikes are

produced quickly after reset, until adaptation builds up

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AdEx: Bursting

[Naud, Marcille, Colpath, Gerstner (2008), Firing patterns in the adaptive exponential integrate-and-fire model]

Red- regular firing Yellow- Adaptive Initial bursting-

Green Blue- Regular

bursting Black- Irregular-

chaotic

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AdEx: Firing Pattern

Figure shows how the choice of reset of voltage (horizontal) and adaptation (vertical) influences firing patterns

Fitting to real Neurons’ data

The parameters of the AdEx model can be fit to match the response of neurons using simple electrophysiological protocols

(current pulses, steps and ramps) AdEx model can reproduce up to 96% of

the spike times of a regular-spiking Hodgkin–Huxley-type model

[Brette and Gerstner, 2005]

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Fitting to real neurons (…contd.)

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With the same set of parameters, AdEx reproduces spikes of the Hodgkin-Huxley model for various firing rates.

Some Limitations

Single-compartment model But works fine

Sodium channel activation is instantaneous In H-H model activation of the sodium current

(via the m variable) is rapid, but lags the evolution of the voltage by a short time in the millisecond range

Downswing of action potential is by resetting to a fixed value after the spike Rapid potassium currents (and also partially by

sodium channel inactivation) is neglected

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Some Limitations (…contd.)

Refractoriness is only represented by the reset of voltage and adaptation variables in real neurons refractoriness is due to

increase in the firing threshold and conductance after a spike and a change in the momentary equilibrium potential

Conductance effects are ignored, because the adaptation variable enters as a current

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REFERENCES

1. Abbott, L.F. (1999). "Lapique's introduction of the integrate-and-fire model neuron (1907)“

2. Izhikevich, E.M. (2001), Resonate-and-fire neurons, Neural Networks, 14:883-894

3. Izhikevich E.M. (2003), Simple model of spiking neurons, IEEE Transactions On Neural Networks, 14:1569-1572

4. Brette R. and Gerstner W. (2005), Adaptive Exponential Integrate-and-Fire Model as an Effective Description of Neuronal Activity, J. Neurophysiol. 94: 3637 - 3642.

5. Naud, Marcille, Colpath, Gerstner (2008), Firing patterns in the adaptive exponential integrate-and-fire model, Biological Cybernetics, DOI 10.1007/s00422-008-0264-7

6. Benda J, Herz A.V.M. (2003), A universal model for spike-frequency adaptation. Neural Comput. 15:2523-2564.

7. http://www.scholarpedia.org/Spike-response_model (doi:10.4249/scholarpedia.1343)

8. Koch, Christof; Idan Segev (1998). Methods in Neuronal Modeling (2 ed.). Cambridge, MA: Massachusetts Institute of Technology. ISBN 0-262-11231-0

9. http://lcn.epfl.ch/~gerstner/SPNM/node26.html

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Thank You!

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Delayed spiking as response of the AdEx model to a current step

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AdEx: Delayed spiking

Voltage as a function of time

[Naud, Marcille, Colpath, Gerstner (2008), Firing patterns in the adaptive exponential integrate-and-fire model]