2. Laguerre Parameterised Hawkes Process

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Neural Coding A Least Squares Approach
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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
brain-machine 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.
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.
3. A Least Squares Approach to Point Process
System Identification 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.
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 We can also
define analogous concepts to auto/cross
covariance, such as the auto intensity (under
stationarity)
(1) (2) (3)
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.
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.
5. Conclusion 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.
2. Laguerre Parameterised Hawkes Process To model
spike train data, we first assume a flexible and
novel parametric form of (1), 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).
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.