Stochastic perturbation methods for spike-timing-dependent plasticity

Constraints on Hebbian and STDP learned weights of a spiking neuron

We analyse mathematically the constraints on weights resulting from Hebbian and STDP learning rules applied to a spiking neuron with weight normalisation. In the case of pure Hebbian learning, we find that the normalised weights equal the promotion probabilities of weights up to correction terms that depend on the learning rate and are usually small. A similar relation can be derived for STDP algorithms, where the normalised weight values reflect a difference between the promotion and demotion probabilities of the weight. These relations are practically useful in that they allow checking for convergence of Hebbian and STDP algorithms. Another application is novelty detection. We demonstrate this using the MNIST dataset.

A model of amygdala function following plastic changes at specific synapses during extinction

The synaptic networks in the amygdala have been the subject of intense interest in recent times, primarily because of the role of this structure in emotion. Fear and its extinction depend on the workings of these networks, with particular interest in extinction because of its potential to ameliorate adverse symptoms associated with post-traumatic stress disorder. Here we place emphasis on the extinction networks revealed by recent techniques, and on the probable plasticity properties of their synaptic connections. We use modules of neurons representing each of the principal components identified as involved in extinction. Each of these modules consists of neural networks, containing specific ratios of excitatory and specialized inhibitory neurons as well as synaptic plasticity mechanisms appropriate for the component of the amygdala they represent. While these models can produce dynamic output, here we concentrate on the equilibrium outputs and do not model the details of the plasticity mechanisms. Pavlovian fear conditioning generates a fear memory in the lateral amygdala module that leads to activation of neurons in the basal nucleus fear module but not in the basal nucleus extinction module. Extinction protocols excite infralimbic medial prefrontal cortex neurons (IL) which in turn excite so-called extinction neurons in the amygdala, leading to the release of endocannabinoids from them and an increase in efficacy of synapses formed by lateral amygdala neurons on them. The model simulations show how such a mechanism could explain experimental observations involving the role of IL as well as endocannabinoids in different temporal phases of extinction.

A unifying theory of synaptic long-term plasticity based on a sparse distribution of synaptic strength

Long-term synaptic plasticity is fundamental to learning and network function. It has been studied under various induction protocols and depends on firing rates, membrane voltage, and precise timing of action potentials. These protocols show different facets of a common underlying mechanism but they are mostly modeled as distinct phenomena. Here, we show that all of these different dependencies can be explained from a single computational principle. The objective is a sparse distribution of excitatory synaptic strength, which may help to reduce metabolic costs associated with synaptic transmission. Based on this objective we derive a stochastic gradient ascent learning rule which is of differential-Hebbian type. It is formulated in biophysical quantities and can be related to current mechanistic theories of synaptic plasticity. The learning rule accounts for experimental findings from all major induction protocols and explains a classic phenomenon of metaplasticity. Furthermore, our model predicts the existence of metaplasticity for spike-timing-dependent plasticity Thus, we provide a theory of long-term synaptic plasticity that unifies different induction protocols and provides a connection between functional and mechanistic levels of description.

Approximating distributions in stochastic learning

On-line machine learning algorithms, many biological spike-timing-dependent plasticity (STDP) learning rules, and stochastic neural dynamics evolve by Markov processes. A complete description of such systems gives the probability densities for the variables. The evolution and equilibrium state of these densities are given by a Chapman-Kolmogorov equation in discrete time, or a master equation in continuous time. These formulations are analytically intractable for most cases of interest, and to make progress a nonlinear Fokker-Planck equation (FPE) is often used in their place. The FPE is limited, and some argue that its application to describe jump processes (such as in these problems) is fundamentally flawed. We develop a well-grounded perturbation expansion that provides approximations for both the density and its moments. The approach is based on the system size expansion in statistical physics (which does not give approximations for the density), but our simple development makes the methods accessible and invites application to diverse problems. We apply the method to calculate the equilibrium distributions for two biologically-observed STDP learning rules and for a simple nonlinear machine-learning problem. In all three examples, we show that our perturbation series provides good agreement with Monte-Carlo simulations in regimes where the FPE breaks down.