Transcript Neural Coding: A Least Squares Approach

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Neural Coding: A Least Squares Approach
Author: Marc Piggott
Supervisor: Professor Victor Solo
Neural Coding
Neural coding is the study of how neurons represent
information. Improved understanding of the neural code has
led to recent developments in neural prostheses and brainmachine interfaces. These devices allow paralysed individuals
to control prosthetic arms or computers (for example), by
interpreting signals from the brain (Fig. 1).
Neurons communicate by sequences of sharp voltage pulses,
called ‘spike trains’. To infer the meaning of spike trains we
perform controlled experiments and develop mathematical
models to describe recorded neural activity (Fig. 2). In this way,
we can classify spike trains as corresponding to particular
movements, sensations or thoughts. This thesis presents a
novel method of system identification for spike train data, to
facilitate such classification.
Output
Underlying Poisson Spiking
spikes/sec
Post -spike filter
Upper: estimated stochastic intensity.
Lower: corresponding spike train
Input
Spikes
Model
Selection
Y
For each
order p
Estimate
, by least
squares
Done?
N
Figure 4. System identification procedure. The true stochastic intensity
is
estimated from an incoming spike train. Upper: Representation of the Hawkes process
with
,
. Lower: Estimation and model selection.
Figure 1. Basic principle of brain-machine
interfaces. (Williams et al. 2009)
Figure 2. UWA cat experiments. A reaching task
is performed to investigate the motor cortex.
(Ghosh et al. 2009)
1. Point Process Modelling
Spike train data consist of distinct events occurring in
continuous time, and therefore cannot be analysed by familiar
techniques. Due to the inherent randomness of spike trains,
we resort to modelling the instantaneous probability rate of
spiking, given the history
(1)
We can also define analogous concepts to auto/cross
covariance, such as the auto intensity (under stationarity)
(2)
Figure 3. “Random” spiking observed from a motor neuron before any
movement occurs. (Ghosh et al. 2009) Occurrence of a spike has no
effect on the occurrence of future spikes apart from the initial
inhibition.
In the point process setting, system identification theory is
under developed. The ubiquitous method is to convert the
spike train to a 0-1 time series and numerically optimize the
appropriate maximum likelihood function. With modern
experiments recording from hundreds of neurons for hours,
such methods are inconvenient. A quicker, and “exact” (no
numerical optimization or discretization) method is desired.
4. Results
We compare our least squares method with that of Pham
(1981) using simulated Hawkes processes. We estimate
directly, whereas Pham bases his estimate of on the estimate
of the expected rate
. Below we present results for
with an average count of 139 spikes,
and for simplicity consider estimating the alphas and kappa
only.
5. Conclusion
2. Laguerre Parameterised Hawkes Process
To model spike train data, we first assume a flexible and novel
parametric form of (1),
(3)
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3. A Least Squares Approach to Point Process
System Identification
where is a constant,
is the impulse response
parameterised by jump parameters
and inverse time
constant , and
is loosely a sequence of delta functions at
the spike times. Observe that (3) may be interpreted as a
filtering operation . The model can be shown to be equivalent
to an all pole filter, see Fig 4 (upper). Note that this is now
partially an impulse response estimation problem (Hawkes,
1971).
In this thesis we developed a novel approach to point process
system identification. Our least squares method was
compared with the only other suggested least squares
approach, and found to out perform in monte carlo simulations
(in terms of mse) for low spike counts. The new approach is
computationally simpler than previous work, taking a more
direct route.
6. Future Work
Derivations of general formulae for the standard errors of our
estimators to further justify observed results. Continued
analysis of the UWA cat experiments, in conjunction with
frequency domain methodology. Extension to bivariate case.
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