Fit Generalised Linear Model to spiking data

A summer project where I wanted to fit a generalised linear model to spiking data and systematically evaluate the performance of a recurrent linear-nonlinear Poisson model. To test the performance of the model data is generated from the Izhikevich neuron (a reduced model of the Hodge-Huxley model).

Generating the data

Data was generated by feeding in random step currents into the Izhikevich dynamical model characterized by

$$ \begin{equation} \begin{cases} \dot{v} = 0.04v^{2} + 5v + 140 − u + I(t)\\ \dot{u} = a(bv − u) \end{cases} \end{equation} $$

$$ \begin{equation} v(t) >= 30 \begin{cases} v (t^{+}) = c \\ u (t^{+}) = u(t) + d \end{cases} \end{equation} $$

This model can describe many ifferent firing behaviour by changing the 4 free parameters $a, b, c$ and $ d$.

The dictionary behaviour_types in Izhikevich.py contains present parameters and current input for the following behaviours:

• Tonic spiking
• Tonic bursting
• Phasic spiking
• Phasic bursting
• Mixed modes
• Spike frequency adaption
• Type I
• Type II
• Spike latency
• Resonator
• Integrator
• Rebound spikes
• Rebound burst
• Threshold variablility
• Bistability I
• Bistability II

This returns a Struct dynamics which characterizes the dynamics, time, and spike events.

From top to bottom: the fast dynamics of the neuron (V), where the dots mark a spiking event, the slow dynamics of the neuron, the input current injected to the neuron, and the binned spike data using a bin width of 500 ms.

Recurrent Linear-Nonlinear Poisson model

Assume spike count, $y$, follows a Poisson Distribution, which is parametrized by a firing rate, $\lambda$. So,

$$ \begin{equation} y_{t} | x_{t} \sim Poiss(\lambda) \end{equation} $$

where the distribution of an event is defined as

$$ \begin{equation} P(y|x, \theta) = \frac{1}{y!}\lambda^{y}e^{-\lambda} . \end{equation} $$

The count rate $\lambda$ is encoded by a linear combination of the stimulus filter $f$ and the history filter $h$. The history filter provides a recurrent feedback to the Poisson spike generator. The stimulus filter describes how the time history of the stimulus affects the spike generator. So theta is a combination $\theta = [f, h]^T$.

To ensure the encoded lambda is positive we pass the encoded stimulus and history through an exponential non-linearity function. Therefore,

$$ \begin{equation} \lambda = f(\theta x) = e^{\theta x}. \end{equation} $$

Building a design matrix

Fitting GLM

You have a set of $Y={y_{i}}$ representing the number of spikes at time $t=i$ and corresponding stimulus and history features $X={x_{i}}$.

To fit the GLM to spiking data we want our parameterised distribution $P_{\theta}(y|x)≈P(y|x)$. This is acheived by finding $\theta$ which maximises our likelihood $P(Y |X, \theta)$.

As each event is assumed to be conditionally independent we can factorise the likelihood probability which yield,

$$ \begin{equation} P(Y|X,\theta) = \prod_{i}^{N} P(y_{i}|x_{i}, \theta) \end{equation} $$

Taking the log of the likelihood will speed up convergence an also reduces the complexity of the analytical solution.

$$ \begin{equation} Log(P(Y|X,\theta)) = \sum_{i}^{N} y_{i}log(\lambda_{i}) -\lambda_{i} -log(y_{t}!) \end{equation} $$

and in the recurrent linear-nonlinear Poisson model,

$$ \begin{equation} \lambda_{i} = e^{x_{i}^{T}\theta} \end{equation} $$

To find the maxima we take the derivative with respect to the $\theta$ which reduces to,

$$ \begin{equation} \partial_{\theta} Log(P(Y|X,\theta)) = YX^{T} - X^{T}e^{X^{T}\theta}. \end{equation} $$

Behaviour and fine tuning

Tonic bursting: bin_width = 2000 n_filter = 17 n_hist_filter = 7

Acknowledgements

Work was a replication of works from on Weber, A. & Pillow, J. 2017

Weber, A. I., & Pillow, J. W. (2017). Capturing the Dynamical Repertoire of Single Neurons with Generalized Linear Models. In Neural Computation (Vol. 29, Issue 12, pp. 3260–3289). MIT Press - Journals. https://doi.org/10.1162/neco_a_01021

Izhikevich, E. M. (2003). Simple model of spiking neurons. In IEEE Transactions on Neural Networks (Vol. 14, Issue 6, pp. 1569–1572). Institute of Electrical and Electronics Engineers (IEEE). https://doi.org/10.1109/tnn.2003.820440