cse/neubeh 528 modeling synapses and networks · cse/neubeh 528 modeling synapses and networks...

1 R. Rao, 528: Lecture 9

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

Linear Recurrent Networks


SYNAPSES!

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


What do synapses do?

Increase or decrease postsynaptic membrane potential

Spike

Image Source: Wikimedia Commons


An Excitatory Synapse

Input spike Neurotransmitter release

(e.g., Glutamate) Binds to receptors Ion channels open

positive ions (e.g. Na+) enter cell

Depolarization due to EPSP (excitatory

postsynaptic potential)


Spike


An Inhibitory Synapse

Input spike Neurotransmitter

release (e.g., GABA) Binds to receptors Ion channels open positive ions (e.g.,

K+) leave cell Hyperpolarization due

to IPSP (inhibitory postsynaptic potential)


Spike


Flashback Membrane Model

lyequivalentor ,)(

A

I

r

EV

dt

dVc e

m

Lm

meLm RIEVdt

dV )(

m = rmcm= RmCm

membrane time

constant

V


Flashback! The Integrate-and-Fire Model

V

meLm RIEVdt

dV )( mV 70LE

If V > Vthreshold Spike

Then reset: V = Vreset

(resting potential)

mV 50thresholdV

Lreset EV

Models a

passive leaky

membrane


Flashback! Hodgkin-Huxley Model

)()()( 3

max,

4

max,max, NaNaKKLLm

emm

EVhmgEVngEVgi

A

Ii

dt

dVc

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

L K Na


How do we model the effects of a synapse on

the membrane potential V ?

Synapse ?


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?

Exponential function K(t) gives reasonable fit to biological data

(other options: difference of exponentials, “alpha” function)

s

t

s

etK

1

)(


Linear Filter Model of Synaptic Input to a Neuron

dtKw

ttKwtI

t

bb

tt

ibb

i

)()(

)()(

Synaptic current for b:

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

Filter for synapse b: s

t

s

etK

1

)(

wb Input Spike

Train b(t)

Synaptic weight


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?


From Spiking to Firing Rate Models

wN

Firing rate ub(t)

Spikes

1(t)

b

t

bb

b

t

bbs

dutKw

dtKwtI

)()(

)()()(

)()( tItIb

bs Total synaptic current

Spike train b(t)

w1 Spikes

N(t)


Synaptic Current Dynamics in Firing Rate Model

F Suppose synaptic kernel 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

)(


Output Firing-Rate Dynamics

F How is the output firing rate v related to synaptic inputs?

F Looks very much like membrane equation:

F On-board derivations of special cases obtained from

comparing the relative magnitudes of r and s …

(see also pages 234-236 in the text)

))(( tIFvdt

dvsr

meLm RIEVdt

dV )(

uw ss

s Idt

dI


How good are Firing Rate Models?

Firing rate model v(t) = F(I(t)) describes this well but not this case

Input I(t) = I0 + I1cos(t)


Feedforward versus Recurrent Networks

)MW( vuvv

Fdt

d

For feedforward networks, matrix M = 0

Output Decay Input Feedback


Example: Linear Feedforward Network

uvv

Wdt

dDynamics:

Steady State (set dv/dt to 0): uv Wss

1

2

2

2

1

10001

11000

01100

00110

00011

10001

W u What is vss?


Linear Feedforward Network

0

1

0

0

1

0

1

2

2

2

1

10001

11000

01100

00110

00011

10001

Wuv ss

What is the network doing?


Linear Filtering for Edge Detection

0

1

0

0

1

0

Output

1

2

2

2

1

Input

W) in versionsshifted (and

00110 Filter

Input

Output


Example of Edge Detection in a 2D Image

http://www.alexandria.nu/ai/blog/entry.asp?E=51


Edge detectors in the visual system

Examples of

receptive

fields in

primary

visual cortex

(V1)

V1

(From Nicholls et al., 1992)


Filtering network is computing derivatives!

)()1(ionapproximat Discrete

)()(lim

0

xfxf

h

xfhxf

dx

df

h

)1()(2)1(

)1()()()1(approx. Disc.

)()(lim

02

2

xfxfxf

xfxfxfxf

h

xfhxf

dx

fd

h

00110

01210


Feedforward Networks: Example 2

Input: Area 7a Neurons with Gaze-Dependent Tuning Curves

Output: Premotor Cortex Neuron with Body-Based Tuning Curves

Coordinate Transformation

(From Section 7.3 in Dayan & Abbott)


Output of Coordinate Transformation Network

Same tuning curve

regardless of gaze angle

Premotor cortex neuron responds

to stimulus location relative to

body, not retinal image location

Head fixed;

gaze shifted to g1 g2 g3

(See section 7.3 in Dayan & Abbott for details)


Linear Recurrent Networks

vuvv

MW dt

d

Output Decay Input Feedback


Next Class: Recurrent Networks

F To Do:

Homework 2

Find a final project topic and partner(s)

cse/neubeh 528 modeling synapses and networks · cse/neubeh 528 modeling synapses and networks...

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