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20 10 School of Electrical Engineering &Telecommunications UNSW ENGINEERING @ UNSW 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). 10 Author: Marc PiggottSupervisor: Professor Victor Solo Neural Coding: A Least Squares Approach 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. 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) 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) (2) (3) Input Spikes For each order p Estimate, by least Done? N Y Model Selection Underlying Poisson Spiking spikes/sec Post -spike filter Output squares Upper: estimated stochastic intensity. Lower: corresponding spike train 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. 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. 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. 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.

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