INTEGRATE-AND-FIRE BASED NEURON MODEL

Presented by –GEETIKA BARMANM.TECH in I.TCSI15006

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A neuron is an electrically excitable cell that processes and transmits information through electrical and chemical signals.

Review Of Neuron

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Neurons can respond to stimuli and conduct impulses because a membrane potential is established across the cell membrane

With one electrode placed inside a neuron & the other outside, the voltmeter reads – 70 mV, which implies that the inside of the neuron is slightly negative relative to the outside.

This difference is referred to as the Resting Membrane Potential.

Two ions contribute: sodium and potassium

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Action Potential• An action potential is a very rapid change in membrane potential that occurs when a nerve cell membrane is stimulated.

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 If a neuron fires then the action potential is the same regardless of the amount of excitation received from the inputs. What is important in neurons is the rate of fire.A weak stimulus will cause a lower rate of fire than a strong stimulus.It has been shown in experiments that the rate of fire of a neuron is directly related to the depolarizing current applied to that neuron.

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Biological Neuron Model :  mathematical description of the properties of neurons , designed to accurately describe and predict biological processes.

Artificial neuron model : aims for computational effectiveness

Neuron Model

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 consists of  an input with some synaptic weight vector. an activation function or transfer function

inside the neuron determining output =f()

Artificial neuron abstraction

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

such as "weight" and "transfer function“. An input to a neuron is an ion current through the cell

membrane described by a physical time dependent current I(t).

an insulating cell membrane that determines a capacitance ().

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

Biological abstraction

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Conductance based neuron model: o biophysical representation of an

excitable cell such as neuron.

oprotein molecule ion channels are represented by conductance and its lipid bilayer by a capacitor.

Integrate and Fire based neuron model

Biological Neuron Model

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Hodgkin and Huxley model

A. Hodgkin and A. Huxley, A Quantitative Description of Membrane Current and Its Application to Conduction and

Excitation in Nerve, JOURNAL OF PHYSIOLOGY, Vol. 117,

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Integrate and Fire Based model

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because of their intrinsic complexity, H.H models are usually difficult to analyze and are computationally expensive in numerical implementations.

Restricts it’s use in describing neural network dynamics.

simple phenomenological spiking neuron models such as integrate-and-fire models are introduced.

used to discuss aspects of neural coding, memory, or network dynamics

WHY I&F MODEL

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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 Lapicque used the model to compute the firing

frequency of a nerve fiber.

Integrate-and-fire model

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In a biologically realistic neural network , a neuron often takes a number of input signals in order to propagate a signal .

Every neuron goes from stable to firing after it reaches a certain threshold. If it fires the signal is passed onto the next neuron, which may or may not fire.

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.

Theoretical Idea

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

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.

Contd…

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A neuron is represented in time by, Which is just the time derivative of the law

of capacitance, Q = CV. When an input current is applied, the membrane

voltage increases with time until it reaches a constant threshold Vth ,at which point a 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..

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The model can be made more accurate. Introduction of a refractory period tref. Limits the firing frequency of a neuron by

preventing it from firing during the refractory period.

The firing frequency as a function of a constant input current thus looks like,

Contd..

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For below-threshold signal, retains the voltage boost forever until it fires again.

Accumulates inputs until the total input passes a threshold.

Not biologically realistic.

Limitation

Leaky Integrate –and-fire model

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

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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.

Leaky Integrate-and-fire model

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Consists of a capacitor C in parallel with a resistor R driven by a current I(t).

Using the law of conservation of current the driving current split into two components

I(t)=IR + IC where IR = U/R ,

IC = dq/dt = C dU/dt

Contd..

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Thus   I(t)= ………..(1) Multiplying eq(1) by R and introducing the

time constant , the above equation becomes

= -U(t) + RI(t)

u membrane potential m membrane time constant

Contd..

…… (2)

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:

Simple variant of I&F model

S.roy, T.Ahmed ,J.Dutta,” A simple variant of Integrate-and Fire model of neuron for application in neuronal area”

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Dendrite is supposed to be consisting of three regions, each receives three inputs from three nearby neurons. Each input is connected with the synaptic weight values to represent the synaptic action.

Effect of all these three inputs is then spatially integrated and brought to a single point value

Each integrated output generates an action potential if it is crosses a threshold value

Assumptions(1)

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The three outputs through the axon are again connected with the synaptic weight values

A comparator integrates these outputs and generates a voltage i.e. membrane voltage.

Action potential is triggered when the membrane voltages reaches a specific threshold value.

Assumptions(2)

28 Fig1: proposed electronic model of neuron

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Fig2:the simulation outputs of dendritic region

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Fig3: the simulated action potential

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http://neuronaldynamics.epfl.ch/online/ http://icwww.epfl.ch/~gerstner/SPNM/ https://en.wikipedia.org/wiki/Biological_neuron_model http://www.stat.columbia.edu/~liam/teaching/neurostat-spr

12/papers/Jolivet04-JNP.pdf http://people.eku.edu/ritchisong/301notes2.htm http://neurotheory.columbia.edu/Larry/AbbottBrResBul99.p

df

References

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Thank You