Network Plasticity as Bayesian Inference

General results from statistical learning theory suggest to understand not only brain computations, but also brain plasticity as probabilistic inference. But a model for that has been missing. We propose that inherently stochastic features of synaptic plasticity and spine motility enable cortical networks of neurons to carry out probabilistic inference by sampling from a posterior distribution of network configurations. This model provides a viable alternative to existing models that propose convergence of parameters to maximum likelihood values. It explains how priors on weight distributions and connection probabilities can be merged optimally with learned experience, how cortical networks can generalize learned information so well to novel experiences, and how they can compensate continuously for unforeseen disturbances of the network. The resulting new theory of network plasticity explains from a functional perspective a number of experimental data on stochastic aspects of synaptic plasticity that previously appeared to be quite puzzling.Comment: 33 pages, 5 figures, the supplement is available on the author's web page http://www.igi.tugraz.at/kappe

Distributed Bayesian Computation and Self-Organized Learning in Sheets of Spiking Neurons with Local Lateral Inhibition

During the last decade, Bayesian probability theory has emerged as a framework in cognitive science and neuroscience for describing perception, reasoning and learning of mammals. However, our understanding of how probabilistic computations could be organized in the brain, and how the observed connectivity structure of cortical microcircuits supports these calculations, is rudimentary at best. In this study, we investigate statistical inference and self-organized learning in a spatially extended spiking network model, that accommodates both local competitive and large-scale associative aspects of neural information processing, under a unified Bayesian account. Specifically, we show how the spiking dynamics of a recurrent network with lateral excitation and local inhibition in response to distributed spiking input, can be understood as sampling from a variational posterior distribution of a well-defined implicit probabilistic model. This interpretation further permits a rigorous analytical treatment of experience-dependent plasticity on the network level. Using machine learning theory, we derive update rules for neuron and synapse parameters which equate with Hebbian synaptic and homeostatic intrinsic plasticity rules in a neural implementation. In computer simulations, we demonstrate that the interplay of these plasticity rules leads to the emergence of probabilistic local experts that form distributed assemblies of similarly tuned cells communicating through lateral excitatory connections. The resulting sparse distributed spike code of a well-adapted network carries compressed information on salient input features combined with prior experience on correlations among them. Our theory predicts that the emergence of such efficient representations benefits from network architectures in which the range of local inhibition matches the spatial extent of pyramidal cells that share common afferent input

Stochastic computations in cortical microcircuit models.

Experimental data from neuroscience suggest that a substantial amount of knowledge is stored in the brain in the form of probability distributions over network states and trajectories of network states. We provide a theoretical foundation for this hypothesis by showing that even very detailed models for cortical microcircuits, with data-based diverse nonlinear neurons and synapses, have a stationary distribution of network states and trajectories of network states to which they converge exponentially fast from any initial state. We demonstrate that this convergence holds in spite of the non-reversibility of the stochastic dynamics of cortical microcircuits. We further show that, in the presence of background network oscillations, separate stationary distributions emerge for different phases of the oscillation, in accordance with experimentally reported phase-specific codes. We complement these theoretical results by computer simulations that investigate resulting computation times for typical probabilistic inference tasks on these internally stored distributions, such as marginalization or marginal maximum-a-posteriori estimation. Furthermore, we show that the inherent stochastic dynamics of generic cortical microcircuits enables them to quickly generate approximate solutions to difficult constraint satisfaction problems, where stored knowledge and current inputs jointly constrain possible solutions. This provides a powerful new computing paradigm for networks of spiking neurons, that also throws new light on how networks of neurons in the brain could carry out complex computational tasks such as prediction, imagination, memory recall and problem solving

Fast convergence of marginals of single neurons and more complex quantities in a cortical microcircuit model.

<p><b>A</b>. Typical spike response of the microcircuit model based on <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1003311#pcbi.1003311-Haeusler1" target="_blank">[30]</a> comprising 560 stochastic point neurons. Spikes of inhibitory neurons are indicated in red. <b>B</b>. Fast convergence of a marginal for a representative layer 5 neuron (frequency of “on”-state, with ) to its stationary value, shown for two different initial Markov states (blue/red). Statistics were obtained for each initial state from trials. <b>C</b>. Gelman-Rubin convergence diagnostic was applied to the marginals of all single neurons (simple states, ). In all neurons the Gelman-Rubin value drops to a value close to 1 within a few , suggesting generally fast convergence of single neuron marginals (shown are 20 randomly chosen neurons; see panel E for a summary of all neurons). The shaded area below 1.1 indicates a range where one commonly assumes that convergence has taken place. <b>D</b>. Convergence speed of pairwise spike coincidences (simple states (1,1) of two neurons, 20 randomly chosen pairs of neurons) is comparable to marginal convergence. <b>E</b>. Summary of marginal convergence analysis for single neurons in C: Mean (solid) and worst (dashed line) marginal convergence of all 560 neurons. Mean/worst convergence is reached after a few . <b>F</b>. Convergence analysis was applied to networks of different sizes (500–5000 neurons). Mean and worst marginal convergence of single neurons are hardly affected by network size. <b>G</b>. Convergence properties of populations of neurons. Dotted: multivariate Gelman-Rubin analysis was applied to a subpopulation of 30 neurons (5 neurons were chosen randomly from each pool). Solid: convergence of a “random readout” neuron which receives spike inputs from 500 randomly chosen neurons in the microcircuit. It turns out that the convergence speed of such a generic readout neuron is even slightly faster than for neurons within the microcircuit (compare with panel E). A remarkable finding is that in all these cases the network size does not affect convergence speed.</p

Solving Sudoku, a constraint satisfaction problem, through structured interactions between stochastically firing excitatory and inhibitory neurons.

<p><b>A</b>. A “hard” Sudoku puzzle with 26 given numbers (left). The solution (right) is defined uniquely by the set of givens and the additional constraints that each digit must appear only once in each row, column and 3×3 subgrid. <b>B</b>. An implementation of the constraints of the Sudoku game in a spiking neural network consists of overlapping WTA circuits. WTA circuits are ubiquitous connection motifs in cortical circuits <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1003311#pcbi.1003311-Douglas1" target="_blank">[29]</a>. A WTA circuit can be modeled by a set of stochastically spiking output neurons that are subject to lateral inhibition (left). The same pyramidal cell can be part of several such WTA motifs (right). In the Sudoku example, each digit in a Sudoku field is associated with four pyramidal cells which vote for this digit when they emit a spike. Each such pyramidal cell participates in four WTA motifs, corresponding to the constraints that only one digit can be active in each Sudoku field, and that a digit can appear only once in each row, column and 3×3 subgrid. <b>C</b>. A typical network run is shown during the last before the correct solution was found to the Sudoku from panel A (the total solve time was approximately in this run, see panel D for statistics of solve times). The network performance (fraction of cells with correct values) over time is shown at the top. The spiking activity is shown for 3 (out of the 81) WTA motifs associated with the 3 colored Sudoku fields in A and B. In each of these WTA motifs there are 36 pyramidal cells (9 digits and 4 pyramidal cells for each digit). Spikes are colored green for those neurons which code for the correct digit in each Sudoku field (6, 8 and 4 in the example). <b>D</b>. Histogram of solve times (the first time the correct solution was found) for the Sudoku from panel A. Statistics were obtained from 1000 independent runs. The sample mean is . <b>E</b>. Average network performance for this Sudoku converges quickly during the first five seconds to a value of , corresponding to % correctly found digits (average taken over 1000 runs; shaded area: standard deviations). Thereafter, from all possible configurations the network spends most time in good approximate solutions. The correct solution occurs particularly often, on average approximately 2% of the time (not shown).</p