CSE/NEUBEH 528 Modeling Synapses and NetworksRecall: Linear Filter Model of a Synapse g K t d g t g...

1R. Rao, 528: Lecture 8

CSE/NEUBEH 528

Modeling Synapses and Networks(Chapter 7)

Image from Wikimedia Commons

Lecture figures are from Dayan & Abbott’s book


Course Summary (thus far)

F Neural Encoding

What makes a neuron fire? (STA, covariance analysis)

Poisson model of spiking

F Neural Decoding

Spike-train based decoding of stimulus

Stimulus Discrimination based on firing rate

Population decoding (Bayesian estimation)

F Single Neuron Models

RC circuit model of membrane

Integrate-and-fire model

Conductance-based Models


Today’s Agenda

F Computation in Networks of Neurons

Modeling synaptic inputs

From spiking to firing-rate based networks

Feedforward Networks

Multilayer Networks


How do neurons

connect to form

networks?

Image Source: Wikimedia Commons

Using

synapses!


Synapses on an actual neuron

Image Credit: Kennedy lab, Caltech. http://www.its.caltech.edu/~mbklab


What do synapses do?

Increase or decrease postsynaptic membrane potential

Spike



An Excitatory Synapse

Input spike Neurotransmitter

release (e.g., Glutamate)

Binds to ion channel receptors

Ion channels open Na+ influx

Depolarization due to EPSP (excitatory

postsynaptic potential)


Spike


An Inhibitory Synapse

Input spike Neurotransmitter

release (e.g., GABA) Binds to ion

channel receptors Ion channels open

Cl- influxHyperpolarization due

to IPSP (inhibitory postsynaptic potential)


Spike


We want a computational model of the effects

of a synapse on the membrane potential V

Synapse

How do we do this?

V


Flashback Membrane Model

,)(

A

I

r

EV

dt

dVc e

m

Lm

meLm RIEVdt

dV )(

m = rmcm= RmCm

is the membrane

time constant

V

or equivalently:

Image Source: Dayan & Abbott textbook


How do we model the effects of a synapse on

the membrane potential V ?

Synapse ?


Hint! Hodgkin-Huxley Model

)()())(/1( 3

max,

4

max, NaNaKKLmm

memmm

EVhmgEVngEVri

RIridt

dV

EL = -54 mV, EK = -77 mV, ENa = +50 mV

K Na

Image Source: Dayan & Abbott textbook


V

srelss

messmLm

PPgg

RIEVgrEVdt

dV

max,

)()(

Probability of transmitter release given an input spike

Probability of postsynaptic channel opening

(= fraction of channels opened)

Synaptic

conductance

Modeling Synaptic Inputs

Synapse


Basic Synapse Model

F Assume Prel = 1

F Model the effect of a single spike input on Ps

F Kinetic Model of postsynaptic channels:

sssss PP

dt

dP )1(

s

s

Opening rate Closing rate

Fraction of channels closed Fraction of channels open

Closed Open

fraction of

channels

opened


What does Ps look like over time given a spike?

Exponential function K(t) gives reasonable fit for some synapses

Others can be fit using “Alpha” function:

s

t

etK

)(

peak

t

ettK

)( t0 peak

Pmax


Linear Filter Model of a Synapse

dtKg

ttKgtg

t

bb

tt

ibb

i

)()(

)()(

max,

max,

Synaptic conductance at b:

b(t) = i δ(t-ti) (ti are the input spike times, δ = delta function)

Filter for

synapse b = )(tK

Input Spike

Train b(t)

Synapse

b


Example: Network of Integrate-and-Fire Neurons

mebbmLm RIEVtgrEVdt

dV ))(()(Each neuron:

Synapses : Alpha function model

Excitatory synapses (Eb = 0 mV) Inhibitory synapses (Eb = -80 mV)

Synchrony!

mV 54 mV 70 threshL VE

ms 01peak


Modeling Networks of Neurons

F Option 1: Use spiking neurons Advantages: Model computation and learning based on:

Spike Timing

Spike Correlations/Synchrony between neurons

Disadvantages: Computationally expensive

F Option 2: Use neurons with firing-rate outputs (real

valued outputs)Advantages: Greater efficiency, scales well to large networks

Disadvantages: Ignores spike timing issues

F Question: How are these two approaches related?


Recall: Linear Filter Model of a Synapse

dtKg

ttKgtg

t

bb

tt

ibb

i

)()(

)()(

max,

max,

Synaptic conductance at b:

b(t) = i δ(t-ti) (ti are the input spike times, δ = delta function)

Filter for

synapse b = )(tK

Synapse b

Input Spike Train b(t)


From a Single Synapse to Multiple Synapses

wN

Spike trains 1(t)

N

b

t

bbs dtKwtI1

)()()(

N

b

bs tItI1

)()(Total synaptic current

w1

N(t)

Synaptic weights


From Spiking to Firing Rate Model

Firing rate ub(t)

Spike train b(t)

N

b

t

bb

N

b

t

bbs

dutKw

dtKwtI

1

1

)()(

)()()(

Firing rate u1(t) uN(t)

wN

Spike trains 1(t)

w1

N(t)

Synaptic weights

Total

synaptic

current


Simplifying the Input Current Equation

Suppose synaptic filter K is exponential:

Differentiating w.r.t. time t,

we get

b

t

bbs dutKwtI )()()(

uw

s

b

bbss

s

I

uwIdt

dI

s

t

s

etK

1

)(

Firing rate u1(t) uN(t)

wNw1Synaptic weights Weight vector w

Input vector u


General Firing-Rate-Based Network Model

))(( tIFvdt

dvsr

uw ss

s Idt

dI

Output firing rate

changes like this:

Input current

changes like this:

What happens when:

input? Static

?

?

sr

rs

F is the “activation function”


Next Class: Networks

F To Do:

Homework 3

Finalize a final project topic and partner(s)

Email Raj, Adrienne and Rich your topic and

partners, or ask to be assigned to a team

CSE/NEUBEH 528 Modeling Synapses and NetworksRecall: Linear Filter Model of a Synapse g K t d g t g...

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