Anthony Bell, Lucas Parra
We use unsupervised probabilistic machine learning ideas to try to ex- plain the kinds of learning observed in real neurons, the goal being to connect abstract principles of self-organisation to known biophysi- cal processes. For example, we would like to explain Spike Timing- Dependent Plasticity (see [5,6] and Figure 3A), in terms of information theory. Starting out, we explore the optimisation of a network sensitiv- ity measure related to maximising the mutual information between input spike timings and output spike timings. Our derivations are analogous to those in ICA, except that the sensitivity of output timings to input tim- ings is maximised, rather than the sensitivity of output ‘ﬁring rates’ to inputs. ICA and related approaches have been successful in explaining the learning of many properties of early visual receptive ﬁelds in rate cod- ing models, and we are hoping for similar gains in understanding of spike coding in networks, and how this is supported, in principled probabilistic ways, by cellular biophysical processes. For now, in our initial simula- tions, we show that our derived rule can learn synaptic weights which can unmix, or demultiplex, mixed spike trains. That is, it can recover inde- pendent point processes embedded in distributed correlated input spike trains, using an adaptive single-layer feedforward spiking network.
1 Maximising Sensitivity.
In this section, we will follow the structure of the ICA derivation  in developing the spiking theory. We cannot claim, as before, that this gives us an information maximisation algorithm, for reasons that we will delay addressing until Section 3. But for now, to ﬁrst develop our approach, we will explore an interim objective function called sensitivity which we deﬁne as the log Jacobian of how input spike timings affect output spike timings.
1.1 How to maximise the effect of one spike timing on another.
Consider a spike in neuron j at time tl that has an effect on the timing of another spike in neuron i at time tk. The neurons are connected by a weight wij. We use i and j to index neurons, and k and l to index spikes, but sometimes for convenience we will use spike indices in place of neuron indices. For example, wkl, the weight between an input spike l and an output spike k, is naturally understood to be just the corresponding wij.