Recording from two neurons: second order stimulus reconstruction from spike trains and population coding

We study the reconstruction of visual stimuli from spike trains, recording simultaneously from the two H1 neurons located in the lobula plate of the fly Chrysomya megacephala. The fly views two types of stimuli, corresponding to rotational and translational displacements. If the reconstructed stimulus is to be represented by a Volterra series and correlations between spikes are to be taken into account, first order expansions are insufficient and we have to go to second order, at least. In this case higher order correlation functions have to be manipulated, whose size may become prohibitively large. We therefore develop a Gaussian-like representation for fourth order correlation functions, which works exceedingly well in the case of the fly. The reconstructions using this Gaussian-like representation are very similar to the reconstructions using the experimental correlation functions. The overall contribution to rotational stimulus reconstruction of the second order kernels - measured by a chi-squared averaged over the whole experiment - is only about 8% of the first order contribution. Yet if we introduce an instant-dependent chi-square to measure the contribution of second order kernels at special events, we observe an up to 100% improvement. As may be expected, for translational stimuli the reconstructions are rather poor. The Gaussian-like representation could be a valuable aid in population coding with large number of neurons.

💡 Research Summary

This paper investigates how visual stimuli can be reconstructed from the spike trains of two simultaneously recorded H1 neurons in the lobula plate of the blowfly Chrysomya megacephala. The authors present a systematic approach that moves beyond simple linear (first‑order) decoding and incorporates second‑order nonlinearities through a Volterra series expansion. The central premise is that, while a first‑order kernel (the linear filter) captures the bulk of the stimulus‑response relationship, the correlations between spikes—especially those arising from simultaneous firing of the two neurons—carry additional information that can only be accessed by including second‑order kernels.

Experimental design
Two H1 neurons, known to encode rotational motion, were extracellularly recorded while the fly was presented with two classes of visual motion: (1) rotational displacements (varying angular velocity) and (2) translational displacements (varying linear velocity). The stimuli were generated as stochastic trajectories (random walks) with time constants ranging from 0.5 s to 2 s, ensuring a rich set of dynamic events. Spike times were digitized at 1 ms resolution, yielding binary spike trains for each neuron.

Volterra decoding framework
The stimulus reconstruction problem is cast as a Volterra series:

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