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What is the neural code? Puchalla et al., 2003

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What is the neural code? Encoding: how does a stimulus cause the pattern of responses? what are the responses and what are their characteristics? neural models: what takes us from stimulus to response; descriptive and mechanistic models, and the relation between them. Decoding: what do these responses tell us about the stimulus? Implies some kind of decoding algorithm How to evaluate how good our algorithm is?

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What is the neural code? Single cells: spike rate spike times spike intervals

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What is the neural code? Single cells: spike rate: what does the firing rate correspond to? spike times: what in the stimulus triggers a spike? spike intervals: can patterns of spikes convey extra information?

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What is the neural code? Populations of cells: population coding correlations between responses synergy and redundancy

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Receptive fields and tuning curves Tuning curve: r = f(s) Gaussian tuning curve of a cortical (V1) neuron

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Receptive fields and tuning curves Tuning curve: r = f(s) Cosine tuning curve of a motor cortical neuron Hand reaching direction

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Receptive fields and tuning curves Sigmoid/logistic tuning curve of a stereo V1 neuron Retinal disparity for a near object

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Reverse correlation Fast modulation of firing by dynamic stimuli Feature extraction Use reverse correlation to decide what each of these spiking events stands for, and so to either: -- predict the time-varying firing rate -- reconstruct the stimulus from the spikes

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Reverse correlation Compute the average stimulus leading up to a spike. Gaussian, white noise stimulus: unbiased stimulus which samples all directions equally S(t) r(t)

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Reverse correlation

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Stimulus = Fluctuating Potential (generates electric field) Spike Train of a Neuron in the ELL of a fish Spike-Triggered Average

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The spike triggered average of the Hodgkin Huxley neuron

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What is the language of single cells? What are the elementary symbols of the code? Most typically, we think about the response as a firing rate, r(t), or a modulated spiking probability, P(r = spike|s(t)). We measure spike times. Implicit: a Poisson model, where spikes are generated randomly with local rate r(t). However, most spike trains are not Poisson (refractoriness, internal dynamics). Fine temporal structure might be meaningful. Consider spike patterns or words, e.g. symbols including multiple spikes and the interval between retinal ganglion cells: when and how much

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Spike Triggered Average 2-Spike Triggered Average (10 ms separation) 2-Spike Triggered Average (5 ms) Multiple spike symbols from the fly motion sensitive neuron

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spike-triggering stimulus feature stimulus X(t) decision function spike output Y(t) x1x1 f1f1 P(spike|x 1 ) x1x1 Decompose the neural computation into a linear stage and a nonlinear stage. Modeling spike generation Given a stimulus, when will the system spike? To what feature in the stimulus is the system sensitive? Gerstner, spike response model; Aguera y Arcas et al. 2001, 2003; Keat et al., 2001 Simple example: the integrate-and-fire neuron

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spike-triggering stimulus feature stimulus X(t) decision function spike output Y(t) x1x1 f1f1 P(spike|x 1 ) x1x1 The decision function is P(spike|x 1 ). Derive from data using Bayes theorem: P(spike|x 1 ) = P(spike) P(x 1 | spike) / P(x 1 ) P(x 1 ) is the prior : the distribution of all projections onto f 1 P(x 1 | spike) is the spike-conditional ensemble : the distribution of all projections onto f 1 given there has been a spike P(spike) is proportional to the mean firing rate Modeling spike generation

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Neuromorphic Engineering

Neuromorphic Engineering

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