Course Summary (thus far) F Neural Encoding What makes a neuron - - PDF document

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• R. 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 2

• R. Rao, 528: Lecture 8

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

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• R. Rao, 528: Lecture 8

Today’s Agenda

F Computation in Networks of Neurons

Modeling synaptic inputs From spiking to firing-rate based networks Feedforward Networks Multilayer Networks

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• R. Rao, 528: Lecture 8

How do neurons connect to form networks?

Image Source: Wikimedia Commons

Using synapses!

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• R. Rao, 528: Lecture 8

Synapses on an actual neuron

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

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• R. Rao, 528: Lecture 8

What do synapses do?

Increase or decrease postsynaptic membrane potential Spike

Image Source: Wikimedia Commons

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• R. Rao, 528: Lecture 8

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)

Image Source: Wikimedia Commons

Spike

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• R. Rao, 528: Lecture 8

An Inhibitory Synapse

Input spike  Neurotransmitter release (e.g., GABA)

 Binds to ion

channel receptors  Ion channels open  Cl- influx Hyperpolarization due to IPSP (inhibitory postsynaptic potential)

Image Source: Wikimedia Commons

Spike

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• R. Rao, 528: Lecture 8

We want a computational model of the effects

• f a synapse on the membrane potential V

Synapse

How do we do this?

V

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• R. Rao, 528: Lecture 8

Flashback Membrane Model

, ) ( A I r E V dt dV c

e m L m

   

m e L m

R I E V dt dV     ) ( 

m = rmcm= RmCm is the membrane time constant

V

• r equivalently:

Image Source: Dayan & Abbott textbook

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• R. Rao, 528: Lecture 8

How do we model the effects of a synapse on the membrane potential V ?

Synapse

?

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• R. Rao, 528: Lecture 8

Hint! Hodgkin-Huxley Model

) ( ) ( ) )( / 1 (

3 max , 4 max , Na Na K K L m m m e m m m

E V h m g E V n g E V r i R I r i dt dV          

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

Image Source: Dayan & Abbott textbook

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• R. Rao, 528: Lecture 8

V

 

s rel s s m e s s m L m

P P g g R I E V g r E V dt 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

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• R. Rao, 528: Lecture 8

Basic Synapse Model

F Assume Prel = 1 F Model the effect of a single spike input on Ps F Kinetic Model of postsynaptic channels:

s s s s s

P P dt dP      ) 1 (

   

s s

 

Opening rate Closing rate Fraction of channels closed Fraction of channels open Closed Open fraction of channels

• pened

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• R. Rao, 528: Lecture 8

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

e t K

 

 ) (

peak

t

e t t K

 

  ) (

t

0 peak Pmax

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• R. Rao, 528: Lecture 8

Linear Filter Model of a Synapse

    d t K g t t K g t g

t b b t t i b b

i

) ( ) ( ) ( ) (

max , max ,

 

  

   

Synaptic conductance at b: b(t) = iδ(t-ti) (ti are the input spike times, δ = delta function) Filter for synapse b =

) (t K

Input Spike Train b(t) Synapse b

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• R. Rao, 528: Lecture 8

Example: Network of Integrate-and-Fire Neurons

m e b b m L m

R I E V t g r E V dt dV       ) )( ( ) ( 

Each neuron: Synapses : Alpha function model

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

Synchrony! mV 54 mV 70    

thresh L

V E

ms 1 

peak



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• R. Rao, 528: Lecture 8

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?

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• R. Rao, 528: Lecture 8

Recall: Linear Filter Model of a Synapse

    d t K g t t K g t g

t b b t t i b b

i

) ( ) ( ) ( ) (

max , max ,

 

  

   

Synaptic conductance at b: b(t) = iδ(t-ti) (ti are the input spike times, δ = delta function) Filter for synapse b =

) (t K

Synapse b Input Spike Train b(t)

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• R. Rao, 528: Lecture 8

From a Single Synapse to Multiple Synapses

wN

Spike trains 1(t)

 

  

 

N b t b b s

d t K w t I

1

) ( ) ( ) (    







N b b s

t I t I

1

) ( ) (

Total synaptic current

w1

N(t) Synaptic weights

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• R. Rao, 528: Lecture 8

From Spiking to Firing Rate Model

Firing rate ub(t) Spike train b(t)

   

     

   

N b t b b N b t b b s

d u t K w d t K w t I

1 1

) ( ) ( ) ( ) ( ) (       

Firing rate u1(t) uN(t)

wN

Spike trains 1(t)

w1

N(t) Synaptic weights Total synaptic current

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• R. Rao, 528: Lecture 8

Simplifying the Input Current Equation

Suppose synaptic filter K is exponential: Differentiating w.r.t. time t, we get

 

 

 

b t b b s

d u t K w t I    ) ( ) ( ) (

u w      



s b b b s s s

I u w I dt dI 

s

t s

e t K







 1 ) ( Firing rate u1(t) uN(t)

wN w1

Synaptic weights Weight vector w Input vector u

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• R. Rao, 528: Lecture 8

General Firing-Rate-Based Network Model

)) ( ( t I F v dt dv

s r

   

u w    

s s s

I dt dI 

Output firing rate changes like this: Input current changes like this: What happens when:

input? Static ? ?

s r r s

     

F is the “activation function”

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• R. Rao, 528: Lecture 8

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