Transcript Modeling working memory and decision making using generic

Rules for Information
Maximization in Spiking Neurons
Using Intrinsic Plasticity
Prashant Joshi & Jochen Triesch
Email: { joshi,triesch }@fias.uni-frankfurt.de | Web: www.fias.uni-frankfurt.de/~{joshi,triesch}
Synopsis

Neurons in various sensory
modalities transform the stimuli into
series of action potentials

The mutual information between
input and output distributions
should be maximized

Biological Evidence: V1 neurons in
cat and macaque respond with an
approximately exponential
distribution of firing rates
Synopsis

Intrinsic plasticity is the persistent modification of a
neuron’s intrinsic electrical properties by neuronal or
synaptic activity

It has been hypothesized that intrinsic plasticity plays a
distinct role in firing rate homeostasis and leads to an
approximately exponential distribution of firing rate

This work derives two gradient based intrinsic plasticity
rules with both rules leading to information maximization

Rule 1: Direct maximization of MI

Rule 2: Minimize the Kullback-Leibler divergence
between OP distribution and a desired exponential
distribution

IP is achieved by adapting the gain function of a neuron to
its input distribution
Outline

Intrinsic plasticity in biology

Computational theory and learning rules
 Neuron model
 IP Rule 1
 IP Rule 2

Simulation Results

Conclusion
Intrinsic Plasticity in Biology

Trace eyelid conditioning
task (Disterhoft et. al. )

Recordings from CA1
pyramidal cells showed a
transient (~1-3 days)
increase in excitability
Figure from: W. Zhang, D. J. Linden. The other side of engram:
Experience-driven changes in neuronal intrinsic excitability. Nat.
Rev. Neurosc., Vol 4, pp. 885-900
Neuron Model
Stochastically spiking neuron with
refractoriness (Toyozumi et. al. )
τm = 10 ms, τrefr = 10 ms, τabs = 3 ms

With refractoriness
Without Refractoriness
IP Rule 1: Direct maximization of mutual information

Key Idea: To maximize
(1)

Equivalent to maximizing
(2)

Where,
(3)

Substituting (3) into (2) we get the term for maximization as:
(4)

Learning rule consist of a set of update equations for various parameters φ of the gain function
(5)
Rule 1 Update Equations
u 0  u 0  u 0
u

u  u
u 0  

MI
u
e
 

g
r0
u  u0  r0
u  
1
e

u 
u
MI
g




Note that similar analysis for the term r0 leads to an update rule which will cause the
value of r0 to increase without any constraint, hence it is not included in the set of
update rules
IP Rule 2: Minimizing the KL Divergence



Key Idea: Minimize the KLD between fy(y) and the optimal exponential distribution fexp(y)
KL Divergence is defined as:
Learning rule consist of a set of update equations for various parameters φ of the gain
function
Rule 2 Update Equations
IP 
r 0  r 0  r 0
g
1  
r 0 
r0   
u 0  u 0  u 0
 
u

u  u
g

 
r 0 
r
0
u 0  1  1  e   1
u   
 
IP
 
g

 
u  u 0   r 0 
r
0
1  1  e   1
u   1 

u 
u    
 
IP
Results: Performance of IP Rule 1 (MI Max)

Inputs: 100 Spike trains,
Gaussian Dist. (mean = 25
Hz. SD = 5 Hz)

ηMI = 10-3, T = 10 min.
u 0  

MI
u
e

g
r0
  u  u 0  rg
u   1 
e
u 
u
MI
0




Results: Performance of IP Rule 2 (KLD Min)

Inputs: 100 Spike trains,
Gaussian Dist. (mean = 30 Hz.
SD = 5 Hz)

ηIP = 10-5, µ= 1.5 Hz, T =
16.67 min.
r 0 
IP 
g
1  
r0 

 
g

 
r 0 
u 0 
1  1  e r 0   1
u 
 
 
IP
g

 
 
u  u 0   r 0 
r
1  1  e   1
u   1 


u 
u    
 
IP
0
Results: Convergence of Rule 2

η

Evolution of trajectories from 3 different initial conditions
IP
= 10-3, µ= 1.5 Hz
Results: Phase Plots

ηIP = 10-3, µ= 1.5 Hz

Pair-wise phase-portraits indicating
the flow field, while keeping the
third parameter constant
Results: Behavior for various IP dist.

η

Gaussian IP Dist: Mean = 30 Hz, SD = 8Hz

Uniform IP Dist: Drawn from [0,60] Hz

Exponential IP Dist: Scale parameter, β = 30 Hz
IP
= 10-3, µ= 1.5 Hz
Results: Adaptation to sensory deprivation

First half: Gaussian
(mean = 30 Hz, SD = 5
Hz)

Second half: Gaussian
(mean = 5 Hz, SD = 1
Hz)
Conclusions

Two simple gradient based rules for IP are presented

First rule used direct maximization of MI

Second rule minimizes the KLD between fy(y) and the optimal exponential distribution
fexp(y)

Adapt the gain function of a model neuron according to sensory stimuli

Valid approach for neuron models which have continuous and differentiable gain functions

Works for several different input distributions

Leads to exponential output distribution, firing rate homeostasis, and adapts to sensory
deprivation
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