CSE/NEUBEH 528 Modeling Synapses and NetworksRecall: Linear Filter Model of a Synapse g K t d g t g...
Transcript of 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
2R. 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
3R. 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
4R. Rao, 528: Lecture 8
How do neurons
connect to form
networks?
Image Source: Wikimedia Commons
Using
synapses!
5R. Rao, 528: Lecture 8
Synapses on an actual neuron
Image Credit: Kennedy lab, Caltech. http://www.its.caltech.edu/~mbklab
6R. Rao, 528: Lecture 8
What do synapses do?
Increase or decrease postsynaptic membrane potential
Spike
Image Source: Wikimedia Commons
7R. 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
8R. Rao, 528: Lecture 8
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)
Image Source: Wikimedia Commons
Spike
9R. Rao, 528: Lecture 8
We want a computational model of the effects
of a synapse on the membrane potential V
Synapse
How do we do this?
V
10R. Rao, 528: Lecture 8
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
11R. Rao, 528: Lecture 8
How do we model the effects of a synapse on
the membrane potential V ?
Synapse ?
12R. Rao, 528: Lecture 8
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
13R. Rao, 528: Lecture 8
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
14R. 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:
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
15R. 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
etK
)(
peak
t
ettK
)( t0 peak
Pmax
16R. Rao, 528: Lecture 8
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
17R. Rao, 528: Lecture 8
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
18R. 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?
19R. Rao, 528: Lecture 8
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)
20R. Rao, 528: Lecture 8
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
21R. Rao, 528: Lecture 8
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
22R. 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
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
23R. Rao, 528: Lecture 8
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”
24R. 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