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Network Plasticity as Bayesian Inference

David Kappel, Stefan Habenschuss, Robert Legenstein, Wolfgang Maass

arXiv:1504.05143v1cs.NEq-bio.NC

TL;DR

How neuronal networks can learn posterior distributions remained an open problem. The paper proposes stochastic plasticity as sampling on a low-dimensional parameter manifold; suitable priors let rewiring respond to input shifts and perturbations while maintaining or restoring function and respecting sparse connectivity.

  • Problem

    How posterior distributions could be learned by networks of neurons remained an open problem.

    Source: §3 · p. 19

  • Method

    The paper proposes stochastic plasticity that samples network parameters from a low-dimensional manifold.

    Source: §3 · p. 19

  • Results

    Suitable priors for spine-volume parameters allow rewiring to respond to input-distribution changes and network perturbations while maintaining or restoring function and observing sparse structural constraints.

    Source: §3 · p. 21

  • Takeaways & Limitations

    The authors conclude that stochasticity may provide an important function for network plasticity by enabling networks to sample parameter configurations.

    Source: §3 · p. 19

    Whether consolidation of a synaptic weight can be adequately modeled as a modification of its prior remains an open question.

    Source: §3 · p. 21

Abstract

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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.

1 Introduction

The paper recasts network plasticity as stochastic sampling from a posterior over configurations, combining structural constraints with fitting to inputs. This offers an alternative to convergence on a single maximum-likelihood solution and is proposed to support generalization and compensation for perturbations.

1 Introduction

  • The framework is presented as addressing overfitting and enabling structural rules, automatic overfitting avoidance, and immediate compensation for network perturbations.The authors demonstrate these demands through three examples of stochastic local plasticity.Source: §1 · pp. 2–3
  • The proposal addresses empirical connection-probability constraints between neuron types and heavy-tailed synaptic-weight distributions, alongside ongoing dendritic-spine and spine-volume fluctuations, including in adult brains and without synaptic activity.Source: §1 · p. 2
  • Structural priors on network parameters combine with the likelihood of inputs to define the posterior, rather than requiring parameters to converge to a maximum-likelihood fit.The paper characterizes the learning goal as high-probability regions of the posterior distribution.Source: §1 · p. 2
  • Stochastic plasticity is proposed to sample network parameters from a posterior that reflects both structural rules and fit to external inputs.The posterior’s high-probability regions define the low-dimensional manifold that serves as the learning goal.Source: §1 · p. 2
  • The proposed learning rule combines activity-dependent fitting, structural constraints, and stochastic drive, allowing parameters to fluctuate between possible solutions.The stochastic term is modeled with a Wiener process, while the fitting and structural terms act as drift.Source: §1 · pp. 2–3
  • The learning theory is developed for unsupervised generative models and applied to non-spiking and spiking networks, integrating structural and synaptic plasticity.The framework relates local plasticity mechanisms to functional network-learning consequences.Source: §1 · p. 3

2 Results

The results develop synaptic sampling as a stochastic framework in which synaptic and structural plasticity sample network configurations from a posterior, combining prior information with ongoing inputs. They describe how this framework supports generalization, network reconfiguration, compensation after perturbations, and changing neural codes that enable cross-modal associations and reconstruction of missing input.

2.1 Learning a posterior distribution through stochastic synaptic plasticity

  • The learning goal combines a prior over network parameters with the likelihood of inputs to define a posterior distribution.The prior can encode structural constraints such as sparse connectivity and rules such as heavy-tailed synaptic weights.Source: §2.1 · p. 4
  • Stochastic local plasticity can make network parameters sample from the posterior distribution as their stationary distribution, rather than converge to a single solution.The posterior is represented by the network’s dynamics, not an explicit neural code.Source: §2.1 · pp. 4–5
  • At low temperature, the dynamics approach deterministic convergence to a local posterior maximum, approximating maximum-a-posteriori inference.The temperature scales the stochastic term; higher values flatten the posterior energy landscape.Source: §2.1 · p. 5

2.2 Online synaptic sampling

  • The online synaptic-sampling rule is a reasonable approximation to batch learning, although theoretical convergence to the correct posterior is not guaranteed.Subsequent simulations use the online rule to demonstrate the viability of this approximation.Source: §2.2 · p. 7

2.3 Relationship to maximum likelihood learning

  • The comparison with maximum-likelihood learning begins from the observation that generative network models typically use gradient-based updates to increase log likelihood.The supplied passage introduces a comparison with synaptic sampling but does not provide its outcome.Source: §2.3 · p. 7
  • Unlike maximum-likelihood optimization, synaptic sampling combines a prior-derived update with ongoing Gaussian noise, enabling continuous sampling from the posterior p*(θ).The noise is not scaled down near a local optimum; the prior term and noise are identified as novel components of the rule.Source: §2.3 · p. 7

2.4 Synaptic sampling improves the generalization capability of a neural network

  • With a uniform weight prior, test performance degrades during prolonged training, whereas a suitable prior maintains good generalization.The example trains on five handwritten 1s; the figure shows the test set contains writing styles absent from training.Source: Figure 2 · p. 8, §2.4 · pp. 9–10
  • In Boltzmann machines, posterior learning uses local plasticity rules and samples both network states and parameters on different timescales.State sampling occurs for given parameters, while parameter sampling proceeds more slowly from the posterior.Source: §2.4 · p. 9

2.5 Spine motility as synaptic sampling

  • Stochastic spine motility can be understood as a special case of synaptic sampling, and the framework permits many spine-motility models.The illustrative model assumes excitatory synapses, multiplicative spine-size dynamics, and one spine per potential connection; the framework can also apply to inhibitory synapses.Source: §2.5 · p. 10
  • A model parameter represents disconnection or functional efficacy; its exponential mapping gives retracted spines near-zero efficacy, and simulated connection survival follows a learning-rate-dependent power law.The survival time scale and power-law exponent depend on the learning rate.Source: §2.5 · p. 10, Figure 3 · p. 11
  • Parameters controlling both connectivity and efficacy unify structural and synaptic plasticity, while a Gaussian parameter prior yields a log-normal prior over synaptic efficacies.Source: §2.5 · p. 12
  • When the likelihood term is small, Eq. (10) implements an Ornstein-Uhlenbeck process, consistent with a previous analysis of spine motility.Source: §2.5 · p. 12
  • Eliminated connections spontaneously regrew at irregular intervals, producing power-law survival that matched new experimental data and qualitatively resembled earlier observations of transient spine decay.Connections that could not contribute to explaining the input disappeared again.Source: §2.5 · p. 12

2.6 Fast adaptation of synaptic connections and weights to a changing input statistics

  • In the WTA model, the likelihood-gradient term gives a simplified STDP rule, and wi=exp(θi−θ0) was reported to fit experimental data better while remaining compatible with the generative model.Source: §2.6 · p. 14
  • Useful synapses survived input-statistics changes while others were eliminated; prior- and noise-driven regrowth became stable when useful, and active synapses remained sparse.The network was exposed to digit 1 examples, then digits 1 and 2, then digit 1 alone.Source: §2.6 · p. 15
  • The model integrates spine motility and STDP in one stochastic rule; its noise supports reconfiguration as input distributions change, while the prior maintains sparse connectivity.The rule is derived from the paper’s stated first-principles goal.Source: §2.6 · p. 15

2.7 Inherent network compensation capability through synaptic sampling

  • The posterior over network parameters is proposed to model functionally equivalent parameter configurations, while continuous stochastic fluctuations can move a perturbed, less-performing network toward a more attractive regime.The authors describe compensation as an inherent feature of synaptic sampling, not an add-on.Source: §2.7 · p. 15
  • Removing visual hidden neurons that significantly encoded stimulus 2 impaired reconstruction, which recovered after about one hour of continued plasticity; removing 2,936 synapses was followed by 294 new hidden-neuron connections after about two hours of sampling.Source: §2.7 · p. 18, Figure 5
  • With visual input absent, visual hidden-neuron activity elicited by auditory digits supported reconstruction of the corresponding visual stimuli.The network formed assemblies encoding digits across auditory and visual populations.Source: §2.7 · p. 18, Figure 5
  • Network parameters fluctuated within a region of parameter space, then switched regions after lesions as visual reconstruction performance recovered.The authors interpret the fluctuations as probing high-probability posterior regions and conclude that ongoing synaptic sampling maintains stable function in a more complex architecture.Source: §2.7 · pp. 18–19
  • Neural assembly sequences for the two digit classes drifted over time, even without lesions, similarly to place-cell and motor-cortex tuning drift and.Source: §2.7 · p. 19

3 Discussion

The discussion frames stochastic synaptic and spine dynamics as posterior sampling that links local plasticity to Bayesian inference, shaping learning, synaptic weights, and network connectivity. It presents ongoing plasticity as supporting adaptation and compensation for perturbations.

3 Discussion

  • Stochasticity lets networks sample low-dimensional manifolds of parameter settings that combine structural constraints with a good fit to empirical evidence.The authors frame high-probability posterior regions as the manifold from which network parameters should be sampled.Source: §3 · p. 19
  • The model predicts that learned stereotypical assembly sequences can shift over longer timescales, but does not specify that timescale, which depends on learning rate.The learning-rate parameter b in Eq. (3) can define the timescale arbitrarily.Source: §3 · pp. 19–20
  • The temperature parameter controls exploration versus exploitation; higher temperature increases spine turnover and enables faster learning and forgetting.With higher temperature, network parameters can move faster to a new posterior peak.Source: §3 · p. 20
  • The Fokker-Planck equation links local stochastic plasticity to the network-level distribution of parameters, grounding rules that approximate Bayesian inference.Those rules allow networks to learn and represent a desired posterior distribution of parameters stochastically.Source: §3 · p. 20
  • The plasticity rules retain a maximum-likelihood term while adding a prior-derived term that can impose structural preferences on network parameters.The prior contribution can differ across biological parameters.Source: §3 · p. 20
  • Suitable priors model dendritic-spine survival statistics and support rewiring that maintains or restores network function after input changes or perturbations.The rewiring also observes structural constraints such as sparse connectivity.Source: §3 · p. 21
  • The framework can also address plasticity in other synaptic parameters, while the authors identify synaptic priors and their possible role in consolidation as questions for further investigation.They suggest that modeling consolidation as a prior modification could support long-term storage alongside flexible responses to later perturbations.Source: §3 · p. 21
  • The discussion addresses the challenge of modeling structural plasticity as Bayesian inference, highlighted by.Source: §3 · p. 21
  • One posterior-based rule for weight changes and rewiring produces persistent synapses and randomly disappearing and reappearing connections, supporting adaptation to changing inputs and network perturbations.The paper relates this rule to functional aspects of the network through the resulting posterior distribution.Source: §3 · p. 21
  • Continuously sampling network parameters from a posterior is proposed to be less brittle than maximum-likelihood learning, because always-active plasticity makes compensation for perturbations automatic.The comparison is with maximum-likelihood frameworks, while the paper's framework keeps plasticity processes active all the time.Source: §3 · p. 21
  • The framework aims to support rethinking neuronal plasticity in light of biological variability and compensation capabilities, a need emphasized by Eve Marder.The article introduces a conceptual and mathematical framework as a foundation for this rethinking.Source: §3 · p. 21

4.1 Details to Learning a posterior distribution through stochastic synaptic plasticity

The section establishes conditions under which stochastic parameter dynamics sample the posterior and develops online, discrete-time, and hidden-state versions of the learning rule.

4.1 Details to Learning a posterior distribution through stochastic synaptic plasticity

  • The posterior p*(θ|x) is the unique stationary distribution of the parameter dynamics, with convergence stated for strictly positive p*.The theorem also states that p* is invariant under the dynamics.Source: §4.1 · p. 22
  • Higher temperature flattens the stationary posterior, while the sampling-speed factor controls exploration and can vary with parameter value without changing that distribution.The temperature scales diffusion, and sampling speed can be reduced for very high or low synaptic weights.Source: §4.1 · p. 22

4.1.1 Online approximation

  • For sufficiently slow sampling, Eq. (18) approximates optimal weight sampling online, addressing the batch rule’s dependence on all inputs and responses.The approximation follows by integrating parameter changes over a full presentation of the data.Source: §4.1.1 · p. 23

4.1.2 Discrete time approximation

  • The discrete-time rule is obtained by integrating the continuous-time rule over Δt, with constant b, T = 1, and η = Δt b.The update is derived over a step during which an input is presented.Source: §4.1.2 · pp. 23–24

4.1.3 Synaptic sampling with hidden states

  • With hidden states, the network samples circuit states and parameters concurrently, yielding a sampling-based version of online expectation maximization.The approach addresses the fact that hidden-variable states are not known from observed data.Source: §4.1.3 · p. 24

4.2 Details to Improving the generalization capability of a neural network through synaptic sampling (Fig. 2)

The section describes an RBM simulation of handwritten digit 1 using synaptic sampling, comparing uniform weight priors with a local bimodal prior.

4.2 Details to Improving the generalization capability of a neural network through synaptic sampling (Fig. 2)

  • A restricted Boltzmann machine with 748 visible and 9 hidden neurons was used to learn distributions over handwritten digit 1 images.The images came from MNIST and were represented as 784-dimensional pixel vectors scaled to [0, 1].Source: §4.2 · p. 24, §4.2.1 · p. 24

4.2.2 Learning procedure

  • Each update used contrastive divergence to estimate likelihood updates, with network outputs after five cycles serving as reconstructions.The procedure alternated visible and hidden neuron updates.Source: §4.2.2 · p. 25
  • Simulations compared learning with an uninformative (uniform) weight prior, implemented by setting ∂/∂w_ij log pS(w) to zero, against factorized local per-weight bimodal priors with means 1.0 and 0.0.Both components had standard deviation 0.15.Source: §4.2.2 · p. 25

4.3 Details to Fast adaptation to changing input statistics (Fig. 4)

This section specifies a stochastic spiking winner-take-all network and its synaptic-sampling rule, then describes simulation settings and MNIST inputs.

4.3.1 Spike-based Winner-Take-All network model

  • The winner-take-all model uses stochastic spike-response neurons whose firing rates depend exponentially on membrane potential, with divisive lateral inhibition.The model also includes a slow homeostatic adaptation current that regulates individual neuron output rates.Source: §4.3.1 · pp. 25–26

4.3.2 Synaptic sampling in spike-based Winner-Take-All networks as stochastic STDP

  • The WTA circuit implements inference over hidden causes: neuron firing rates are proportional to posterior probabilities, and synaptic sampling supplies stochastic STDP.The generative model assumes inputs are conditionally distributed given a hidden cause.Source: §4.3.2 · p. 27

4.3.3 Simulation details for spiking network simulations (Figs. 3, 4, and 5)

  • In all spiking neural network simulations, synaptic weights were updated according to rule (11), with N = 100, α = e^-2, and b = 10^-4, except panel 3F, where b = 10^-6 was used as a control.Source: §4.3.3 · p. 28

4.3.4 Network inputs for Figs. 3 and 4

  • For Figs. 3 and 4, MNIST pixels drove afferent neurons with 0–50 Hz Poisson rates plus 1 Hz noise, fixed for 200 ms per digit.A 50 ms interval of 1 Hz noise separated digit presentations.Source: §4.3.4 · p. 28

4.4 Details to Inherent compensation capabilities of networks with synaptic sampling (Fig. 5)

The section describes recurrent winner-take-all networks trained with a shared synaptic-sampling rule and paired spoken-digit and handwritten-digit inputs.

4.4.1 Network model

  • The model used two recurrently connected ensembles, each containing four WTA circuits, with the same learning rule applied to lateral and feedforward synapses.Lateral connections were unconstrained and had a 5 ms synaptic delay.Source: §4.4.1 · p. 28

4.4.2 Network inputs

  • Training paired spoken digit utterances with corresponding handwritten digits; the speech inputs were reconstructed cochleagrams from TI 46.The inputs used 77 cochleagram channels represented by 10 afferent neurons each, with 7 of 10 utterances used for training and 3 for testing.Source: §4.4.2 · pp. 28–29
  • For auditory-only test trials, visual input neurons were held at 1 Hz and learning was disabled.PETH plots summarized 100 network responses to the same stimulus class.Source: §4.4.2 · p. 29

5 Supplement

The supplement is available on the author's web page.

5 Supplement

  • The supplement is available on the author's web page.http://www.igi.tugraz.at/kappel/Source: §5 · p. 29
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