Ron Meir

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State Estimation and Prediction Based On Dynamic
Spike Train Decoding Noise, Adaptation, and
Multimodality
Ron Meir Department of Electrical
Engineering Technion, Israel
With Omer Bobrowski and Yonina Eldar
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The Problem
Noise ubiquitous
Sensory
Processing
Environment
World state
State estimator
Objective Based on partial noisy sensory input
- estimate world state
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Desiderata

• Compute posterior distribution
• P(current statesensory input) or
• P(next statesensory input)
• Continuous time
• Online - process each spike upon arrival
• Real time fixed computational load
• Implementation by a recurrent neural network

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Summary of Main Results

• Main assumption State is a Markov process
• Mathematical foundations developed in the 70s
• Suggest implementation by a simple bi-linear
  neural network
• Extend the original framework
• Noisy input
• History dependent spike trains (exploring
  adaptation)
• Multimodal inputs Prediction
• Demonstrate known results from static-case

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Comparative Dimensions
Main restrictions (1) finite-state (2) Markovian
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Problem Setup

• - A process representing world state
• - Sensory cell
  responses
• Spike trains
• Partial, noisy, delayed, redundant,

..
?
. . .
.. . .
Decoding Network
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Problem Formulation

• Main assumptions
• World state finite-state continuous-time Markov
  process
  with a generator matrix
• Sensory activity Poisson processes with rates
  - tuning curves
• Objective
• Compute posterior probabilities
• Requirements
• Online real time computation
• Neural network implementation

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Mathematical Framework Nonlinear Filtering

• History
• Formulated in sixties (Kushner, Zakai, Wonham)
• Extended in seventies to point processobservation
  s (Snyder, Segal, Kailath)
• Our work
• Relies on this rigorous theory
• Demonstrates real-time neural implementation
• Extensions Generalizations multimodality,
  noise, non-Poisson,

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Key Concept

• Zakai Equation
• A simple filter for the non-normalized
  probabilities
• Meaning
• There is a special set of functions -Such
  that
• Computing probabilities hardComputing
  non-normalized probabilities easy

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Filtering Equation
Stochastic Differential Equation
prior
sensory data
no spikes bias
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Neural Implementation
Sensory layer
Posterior network
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Example Visual Tracking
No input Increased uncertainty
Objects trajectory
0
1
Probability
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System Behavior

• Between spikes

• Spike arrival from the mth sensory cell

• Closed form solution available

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The Static Case

• Setup
• Constant input, selected from finite set
• Assumptions
• Gaussian prior
• Gaussian tuning-curves
• Posterior
• Gaussian -

flat prior
Population vector(Georgopoulos 82)
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Generalization and Extensions
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Environmental Noise

• Previous setup
• Noisy setup
• Rates are functions of the noisy state

Limitation Sensory response is a direct function
of the state
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Environmental Noise

• Assumptions
• - a finite-state Markov process (
  )
• are independent
• Tuning-curves
• E.g., additive noise -
• Solution
• Key observation is a
  Markov process ( )
• Write recursive equation computing
• Compute the marginal non-normalized probabilities

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Environmental Noise

• Result

where
Average sensory response (with regard to the
noise)
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Environmental Noise
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Optimal Tuning Curve Width

• Motivation

C High precision D Poor coverage D Few spikes
Expected optimal tuning curve width

• D Low precision
• C Good coverage
• C Many spikes

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Optimal Tuning Curve Width

• Setup
• Static stimulus
• Additive noise
• Assumptions
• Gaussian prior
• Gaussian noise
• Gaussian tuning-curves
• Optimality criterion
• MSE of the optimal estimator -

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History Dependent Spike Trains

• Why not Poisson?
• Lack memory
• Physiological phenomena
• Refractory period firing is exhausting
• Adaptation neurons get bored

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History Dependent Spike Trains

• Self-exciting point processes
• Rate depends on history
• Includes Poisson, Renewals, and more
• Equation

(Synaptic depression ?)
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History Dependent Spike Trains

• Using spikes efficiently

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History Dependent Spike Trains

• Self inhibition term
• No adaptation
• With adaptation

cells near the true state are the least inhibited
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Multisensory Integration

• Setup
• Two or more input modalities (e.g. visual and
  auditory)
• Goal
• Compute posteriorbased on both modalities

A
quack
V
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Multisensory Integration

• Reminder unimodal equation
• Multimodal equation
• where
• - number of visual/auditory sensory cells
• - mth visual/auditory input activity
• - mth visual/auditory cells tuning-curve

multi-sensory data
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Neural Implementation
Sensory layer
Posterior network
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Multisensory Integration

• Possible questionIs it the same as taking
  inputs of the same modality?
• Answer no
• Biologically
• Different information conveyed by different
  tuning-curve properties (shape, latency, )
• Mathematically
• Different noise processes
• Benefit - two noisy observations instead of one

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Multisensory Integration

• Example
• Auditory input compensates for the lack of visual
  input
• And vice versa

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Multisensory Integration

• Comparing multimodal vs. unimodal computation
• - constant
• - varying
• Multimodal inputs of one modality and
  of another
• Unimodal inputs of the same
  modality

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Multisensory Integration The static case

• Setup Constant input, multimodal observations
• Well studied case Deneve, Latham, Pouget, others
• Assumption Gaussian unimodal posteriors
• Result Gaussian multimodal posterior

visual posterior network
auditory posterior network
image
sound
image
multimodal posterior network
sound
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Multisensory Integration The static case

• In our framework
• Assumptions Gaussians tuning-curves and prior
  (as in the unimodal case)
• ResultGaussian multimodal posterior, with

flat prior
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Prediction

• Objective
• Compute posterior distribution of future states
• Solution
• Define (non-normalized probabilities of
  future state)
• Then (when transition-matrix is
  regular)
• Substitute into the original equation to get

where
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Summary

• Spike based filtering of a continuous-time Markov
  process
• Online implementation using bi-linear neural
  networks
• Mathematically rigorous treatment of continuous
  time
• No temporal information lost
• Multimodal effects easily accounted for
• Many extensions
• noise, history-dependent spike trains,
  prediction, and more
• Demonstrating static-case known results

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Extensions

• Learning and adaptation
• Distributed and robust representation
• Continuous state-space
• Physiological interpretation and implementation
• Biological experiments

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Thanks!
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Prediction
Sensory layer
Posterior network
Prediction layer
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Recent Work Overview

• Pitkow, Sompolinsky Meister (PLoS Biology,
  2007)
• Setup
• Static stimulus horizontal/vertical bar
• Dynamic noise fixational eye movements (2D
  random walk, continuous time, discrete space)
• Sensory activity independent Poisson spike
  trains
• Firing rate represents probability
• Purpose
• Determine objects retinal position
• Distinguish between horizontal/vertical bars
• Computation by a rate-based recurrent neural
  network

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Recent Work Overview

• Pitkow, Sompolinsky Meister (PLoS Biology,
  2007)
• Dynamic posterior update

stimulus S at position x, given spiking activity
up to time t
spike train generated by retinal neuron y
tuning-curve related functions
discrete-space differential
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Recent Work Overview

• Pitkow, Sompolinsky Meister (PLoS Biology, 2007)

Pitkow et al. (PLoS Biology, 2007)
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Recent Work Overview

• Pitkow, Sompolinsky Meister (PLoS Biology,
  2007)
• Summary
• Equation neural implementation similar to our
  work
• Provides evidence for biological plausibility in
  V1
• Compares to psychophysical experiments
• Comparing to this work
• Handles 2D random walks only
• Does not go beyond the basic equation (e.g.
  noise, prediction, multisensory, adaptation, etc.)

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Recent Work Overview

• Deneve (Neural Computation, 2008)
• Setup
• Dynamic stimulus a continuous-time binary
  Markov process
• Sensory activity Poisson spike trains
• Neurons encode probability as an inner state
• Purpose
• Compute the log-likelihood ratio
• Represent the ratio in a single cell activity

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Recent Work Overview

• Deneve (Neural Computation, 2008)
• Summary
• A differential equation for computing the
  log-ratio
• A single cell mechanism to propagate probability
• Providing learning mechanisms
• Comparing to this work
• Binary stimulus only
• Nonlinear equation
• Inner state (posterior distribution) computation
  mechanism is not studied
• Neural implementation might lose information
• The log-ratio equation is a special case of
  framework suggested in our work

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Recent Work Overview

• Beck Pouget (Neural Computation, 2007)
• Setup
• Dynamic stimulus a continuous-time Markov chain
• Sensory activity abstract
• Firing rate represents probability
• Purpose
• Determine the stimulus state from sensory input
• Computation by a rate-based recurrent neural
  network

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Recent Work Overview

• Beck Pouget (Neural Computation, 2007)

posterior state distribution, given spiking
activity up to time t
stimulus generator matrix
unspecified functions of the input spikes
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Recent Work Overview

• Beck Pouget (Neural Computation, 2007)
• Summary
• Posterior calculation by a recurrent neural
  network
• Comparing to this work
• Approximated derivation
• Quadratic equation
• Unspecified input terms

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Recent Work Overview

• Rao (Neural Computation 2004, NIPS 2005)
• Setup
• Dynamic stimulus a discrete-time Markov chain
• Sensory activity abstract
• Firing rate represents probability
• Purpose
• Determine the stimulus state from sensory input
• Computation by a rate-based recurrent neural
  network

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Recent Work Overview

• Rao (Neural Computation 2004, NIPS 2005)
• Summary
• Posterior calculation by a discrete-time
  recurrent neural network
• General structure similar to previous models
• Comparing to this work
• Discrete time
• Approximated derivations (log of sum, random
  spikes)
• Nonlinear interactions
• Unspecified input terms

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Recent Work Overview

• Brown, Barbieri, Solo Co.
• Setup
• Dynamic stimulus a discrete-time Gaussian AR
  models.
• Sensory activity independent Poisson spike
  trains
• Assumed receptive fields Gaussian/Zernekie
  polynomials
• Purpose
• Computerized decoding of information from neural
  activity (Kalman based approach)
• Highly different purpose than our work implies
  different requirements/assumptions/approximations

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Recent Work Overview

• Brown, Barbieri, Solo Co.
• Main differences with our work
• Implementation
• computer vs. neural network
• discrete vs. continuous time
• Nonlinear equations, required iterative solutions
  vs.simple bi-linear equations
• Model assumptions
• discrete-time, continuous-space (Gaussian AR) vs.
  continuous-time, discrete-space (finite-state
  Markov)
• special vs. generic receptive fields
• Gaussian approximation for the posterior vs. no
  approximation

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Example Multi-target Tracking

• Combined state-space
• Marginal probabilities