Past exam of the mathematics course of the University of Cambridge / 2007 / iii / Paper 73 / 1

An action potential is a brief regenerative change in a neuron's membrane potential, followed by repolarization and recovery. In a typical spiking neuron, depolarization recruits voltage-gated ion channels; inward current promotes further depolarization until inactivation and opposing currents terminate the event. The neuronal refractory period makes immediate repetition difficult or impossible. Once initiated, the roughly stereotyped event propagates along the axon. Its amplitude is therefore not generally the primary variable used to represent stimulus strength.

Inputs arriving at synapses on a neuron and its dendrites change membrane currents and produce graded voltage changes. Their spatial and temporal integration can trigger an action potential. At a chemical synapse, an arriving spike promotes release of a neurotransmitter, whose receptors alter the next neuron's conductances. Depending on the synapse and receptor mechanism, the input can promote or suppress the next neuron's firing. Thus information flows through activity patterns interacting with connectivity and synaptic dynamics, rather than by an identical voltage waveform simply passing unchanged across every cell.

In rate coding, a relevant response statistic is the number of spikes in a chosen interval, or the mean firing rate inferred from repeated trials or a population. For example, a sensory variable might be represented by mean counts of two versus five spikes in a $100$ ms window. A downstream neuron integrating many synaptic inputs over that window can distinguish the rates even if individual spike times vary. The window, averaging method and neuron's stimulus tuning must be specified; not every cell increases its rate for every stronger stimulus. Neural adaptation can also make the rate depend on recent stimulus history.

In temporal coding, precise times carry information beyond that count. Two responses containing the same number of spikes can differ in their first-spike latency after stimulus onset. Alternatively, spikes can lock to a particular phase of a periodic input, or relative timing across two neurons can encode coincidence. A decoder that reads only the long-window count discards those distinctions, whereas a temporally sensitive synapse or coincidence detector can use them.

Rate coding uses a chosen frequency or count summary; temporal coding uses the arrangement of spike times relative to a reference. They are complementary possibilities within neural coding, and a spike train can carry information in both. Stochastic variability does not prevent coding, but requires assessing how reliably different response distributions can be distinguished.

For a constant-rate Poisson process, let $T$ be the waiting time from a spike to the next spike. By independent increments, there is no spike in an interval of length $u$ with probability $e^{-ru}$. Hence, for $r>0$,Its density is $r{e}^{-ru}$, so the interspike intervals are independent exponential distributions with mean $1/r$. If $U$ is uniform on $(0,1)$, inverse transform sampling gives $T=-\log U/r$. Repeatedly add independent draws to the current spike time to simulate a constant-rate spike train.

For a prescribed time-dependent nonnegative rate $r(t)$, use an Inhomogeneous Poisson process. Conditional on the present event at $t_i$, its next waiting time has survival probabilityThus its conditional CDF is $1$ minus that expression for $u\ge 0$. Define the cumulative intensity $\Lambda (t)={\int }_0^{t}r(s)ds$. For each new independent uniform draw, set $E_i=-\log U_i$ and solveThis transforms independent unit-rate exponential intervals in integrated-intensity time into physical spike times. Use a generalized inverse when the rate vanishes on an interval: no spikes are placed in a flat part of $\Lambda$. On a finite simulation window, stop when the next intensity target exceeds the available $\Lambda$. If the total future intensity is finite, a target beyond it means that no further spike occurs.

Numerically, integrate the known rate and invert the accumulated integral, or integrate exactly across piecewise-constant rate bins. Drawing an exponential interval using only $r(t_i)$ is not exact when the rate changes during the interval. The Poisson model also omits the neuronal refractory period and other spike-history effects.

One useful starting point is reverse correlation, implemented as a spike-triggered average. Present a long stimulus record and retain the stimulus history before each observed spike. After subtracting the stimulus mean, estimatewhere $K$ is the number of spikes and $T_k$ is the history window. This identifies a candidate temporal receptive field. For white Gaussian stimulation and a one-filter encoding model, the average is proportional to the relevant filter, provided the response nonlinearity does not make that average vanish. For example an even response to the projection can have zero average despite strong stimulus dependence. Colored inputs require accounting for stimulus covariance, and more than one relevant stimulus direction may require a richer fitted model.

Use the candidate filter in an LNP model:Estimate the static nonlinearity $g$ by binning filter projections and dividing spike counts by exposure time in each bin. Equivalently, Bayes' rule gives $g(z)=\overline{r}p(z\mid \text{spike})/p(z)$ under this model. Refine the filter and nonlinearity by maximizing the conditional Poisson log-likelihoodwith suitable regularization. Simulate an Inhomogeneous Poisson process at the fitted rate using integrated-intensity inversion from the previous solution. This gives a statistical encoding model, not a claim to reconstruct the neuron's ion channels.

Use cross-validation: estimate parameters on one set of trials and test them on independent held-out trials. Under repeated identical stimuli, compare the observed and simulated trial-averaged firing rates, spike-count distributions, interspike-interval distributions and autocorrelations. Compare held-out likelihoods and predictive errors with simpler baseline models and with the variability between real repeats. Exact spike-by-spike agreement is not expected from two independent realizations of a stochastic model. Large rate agreement accompanied by incorrect short-interval or correlation statistics reveals missing dynamics. Adding a spike-history term, for example in a point-process generalized linear model, can account for refractoriness or bursting when the basic Poisson cascade fails.

A good model predicts held-out response distributions, not just its training spike train. Such a model summarizes what stimulus features affect the neuron, permits controlled simulated changes of those features, and tests whether the proposed computation is sufficient. Its failures can suggest further experiments or additional stimulus and history variables.

Assume the two response labels are distinct and the error probability is the same for each stimulus. Then the stimulus-response map is a binary symmetric channel:with the two incorrect conditional probabilities equal to $P_x$. Equal stimulus probabilities give $\mathbb{P}(R=r_{+})=\mathbb{P}(R=r_{-})=1/2$. In bits, the Shannon entropy of the response is therefore $H(R)=1$. For either stimulus the response entropy is the binary entropywith $0\log 0=0$. Thus $H_{\mathrm{noise}}=H(R\mid S)=h_2(P_x)$, and the mutual information isIt is maximal, equal to one bit, at $P_x=0$ and also at $P_x=1$. The latter response is always opposite to the ideal label but still specifies the stimulus perfectly after relabelling. It is minimal, equal to zero, at $P_x=1/2$, when both response distributions are identical and the response is independent of the stimulus. If error is conventionally restricted to $0\le P_x\le 1/2$, the maximum occurs only at zero.

The symmetric-error assumption matters. A single overall error probability would not determine this information if the two stimuli had different conditional error rates or if the response included additional outcomes.