Memory nearly on a spring: a mean first passage time approach to memory lifetimes

The Impact of Sparse Coding on Memory Lifetimes in Simple and Complex Models of Synaptic Plasticity

Models of associative memory with discrete state synapses learn new memories by forgetting old ones. In the simplest models, memories are forgotten exponentially quickly. Sparse population coding ameliorates this problem, as do complex models of synaptic plasticity that posit internal synaptic states, giving rise to synaptic metaplasticity. We examine memory lifetimes in both simple and complex models of synaptic plasticity with sparse coding. We consider our own integrative, filter-based model of synaptic plasticity, and examine the cascade and serial synapse models for comparison. We explore memory lifetimes at both the single-neuron and the population level, allowing for spontaneous activity. Memory lifetimes are defined using either a signal-to-noise ratio (SNR) approach or a first passage time (FPT) method, although we use the latter only for simple models at the single-neuron level. All studied models exhibit a decrease in the optimal single-neuron SNR memory lifetime, optimised with respect to sparseness, as the probability of synaptic updates decreases or, equivalently, as synaptic complexity increases. This holds regardless of spontaneous activity levels. In contrast, at the population level, even a low but nonzero level of spontaneous activity is critical in facilitating an increase in optimal SNR memory lifetimes with increasing synaptic complexity, but only in filter and serial models. However, SNR memory lifetimes are valid only in an asymptotic regime in which a mean field approximation is valid. By considering FPT memory lifetimes, we find that this asymptotic regime is not satisfied for very sparse coding, violating the conditions for the optimisation of single-perceptron SNR memory lifetimes with respect to sparseness. Similar violations are also expected for complex models of synaptic plasticity.

A Mathematical Analysis of Memory Lifetime in a Simple Network Model of Memory

We study the learning of an external signal by a neural network and the time to forget it when this network is submitted to noise. The presentation of an external stimulus to the recurrent network of binary neurons may change the state of the synapses. Multiple presentations of a unique signal lead to its learning. Then, during the forgetting time, the presentation of other signals (noise) may also modify the synaptic weights. We construct an estimator of the initial signal using the synaptic currents and in this way define a probability of error. In our model, these synaptic currents evolve as Markov chains. We study the dynamics of these Markov chains and obtain a lower bound on the number of external stimuli that the network can receive before the initial signal is considered forgotten (probability of error above a given threshold). Our results are based on a finite-time analysis rather than large-time asymptotic. We finally present numerical illustrations of our results.

First Passage Time Memory Lifetimes for Multistate, Filter-Based Synapses

Models of associative memory with discrete state synapses learn new memories by forgetting old ones. In contrast to non-integrative models of synaptic plasticity, models with integrative, filter-based synapses exhibit an initial rise in the fidelity of recall of stored memories. This rise to a peak is driven by a transient process and is then followed by a return to equilibrium. In a series of papers, we have employed a first passage time (FPT) approach to define and study memory lifetimes, incrementally developing our methods, from both simple and complex binary-strength synapses to simple multistate synapses. Here, we complete this work by analyzing FPT memory lifetimes in multistate, filter-based synapses. To achieve this, we integrate out the internal filter states so that we can work with transitions only in synaptic strength. We then generalize results on polysynaptic generating functions from binary strength to multistate synapses, allowing us to examine the dynamics of synaptic strength changes in an ensemble of synapses rather than just a single synapse. To derive analytical results for FPT memory lifetimes, we partition the synaptic dynamics into two distinct phases: the first, pre-peak phase studied with a drift-only approximation, and the second, post-peak phase studied with approximations to the full strength transition probabilities. These approximations capture the underlying dynamics very well, as demonstrated by the extremely good agreement between results obtained by simulating our model and results obtained from the Fokker-Planck or integral equation approaches to FPT processes.

Dynamic Integrative Synaptic Plasticity Explains the Spacing Effect in the Transition from Short- to Long-Term Memory

Repeated stimuli that are spaced apart in time promote the transition from short- to long-term memory, while massing repetitions together does not. Previously, we showed that a model of integrative synaptic plasticity, in which plasticity induction signals are integrated by a low-pass filter before plasticity is expressed, gives rise to a natural timescale at which to repeat stimuli, hinting at a partial account of this spacing effect. The account was only partial because the important role of neuromodulation was not considered. We now show that by extending the model to allow dynamic integrative synaptic plasticity, the model permits synapses to robustly discriminate between spaced and massed repetition protocols, suppressing the response to massed stimuli while maintaining that to spaced stimuli. This is achieved by dynamically coupling the filter decay rate to neuromodulatory signaling in a very simple model of the signaling cascades downstream from cAMP production. In particular, the model's parameters may be interpreted as corresponding to the duration and amplitude of the waves of activity in the MAPK pathway. We identify choices of parameters and repetition times for stimuli in this model that optimize the ability of synapses to discriminate between spaced and massed repetition protocols. The model is very robust to reasonable changes around these optimal parameters and times, but for large changes in parameters, the model predicts that massed and spaced stimuli cannot be distinguished or that the responses to both patterns are suppressed. A model of dynamic integrative synaptic plasticity therefore explains the spacing effect under normal conditions and also predicts its breakdown under abnormal conditions.

First Passage Time Memory Lifetimes for Simple, Multistate Synapses: Beyond the Eigenvector Requirement

Models of associative memory with discrete-strength synapses are palimpsests, learning new memories by forgetting old ones. Memory lifetimes can be defined by the mean first passage time (MFPT) for a perceptron's activation to fall below firing threshold. By imposing the condition that the vector of possible strengths available to a synapse is a left eigenvector of the stochastic matrix governing transitions in strength, we previously derived results for MFPTs and first passage time (FPT) distributions in models with simple, multistate synapses. This condition permits jump moments to be computed via a 1-dimensional Fokker-Planck approach. Here, we study memory lifetimes in the absence of this condition. To do so, we must introduce additional variables, including the perceptron activation, that parameterize synaptic configurations, permitting Markovian dynamics in these variables to be formulated. FPT problems in these variables require solving multidimensional partial differential or integral equations. However, the FPT dynamics can be analytically well approximated by focusing on the slowest eigenmode in this higher-dimensional space. We may also obtain a much better approximation by restricting to the two dominant variables in this space, the restriction making numerical methods tractable. Analytical and numerical methods are in excellent agreement with simulation data, validating our methods. These methods prepare the ground for the study of FPT memory lifetimes with complex rather than simple, multistate synapses.

First Passage Time Memory Lifetimes for Simple, Multistate Synapses

Memory models based on synapses with discrete and bounded strengths store new memories by forgetting old ones. Memory lifetimes in such memory systems may be defined in a variety of ways. A mean first passage time (MFPT) definition overcomes much of the arbitrariness and many of the problems associated with the more usual signal-to-noise ratio (SNR) definition. We have previously computed MFPT lifetimes for simple, binary-strength synapses that lack internal, plasticity-related states. In simulation we have also seen that for multistate synapses, optimality conditions based on SNR lifetimes are absent with MFPT lifetimes, suggesting that such conditions may be artifactual. Here we extend our earlier work by computing the entire first passage time (FPT) distribution for simple, multistate synapses, from which all statistics, including the MFPT lifetime, may be extracted. For this, we develop a Fokker-Planck equation using the jump moments for perceptron activation. Two models are considered that satisfy a particular eigenvector condition that this approach requires. In these models, MFPT lifetimes do not exhibit optimality conditions, while in one but not the other, SNR lifetimes do exhibit optimality. Thus, not only are such optimality conditions artifacts of the SNR approach, but they are also strongly model dependent. By examining the variance in the FPT distribution, we may identify regions in which memory storage is subject to high variability, although MFPT lifetimes are nevertheless robustly positive. In such regions, SNR lifetimes are typically (defined to be) zero. FPT-defined memory lifetimes therefore provide an analytically superior approach and also have the virtue of being directly related to a neuron's firing properties.

Mean First Passage Memory Lifetimes by Reducing Complex Synapses to Simple Synapses

Memory models that store new memories by forgetting old ones have memory lifetimes that are rather short and grow only logarithmically in the number of synapses. Attempts to overcome these deficits include "complex" models of synaptic plasticity in which synapses possess internal states governing the expression of synaptic plasticity. Integrate-and-express, filter-based models of synaptic plasticity propose that synapses act as low-pass filters, integrating plasticity induction signals before expressing synaptic plasticity. Such mechanisms enhance memory lifetimes, leading to an initial rise in the memory signal that is in radical contrast to other related, but nonintegrative, memory models. Because of the complexity of models with internal synaptic states, however, their dynamics can be more difficult to extract compared to "simple" models that lack internal states. Here, we show that by focusing only on processes that lead to changes in synaptic strength, we can integrate out internal synaptic states and effectively reduce complex synapses to simple synapses. For binary-strength synapses, these simplified dynamics then allow us to work directly in the transitions in perceptron activation induced by memory storage rather than in the underlying transitions in synaptic configurations. This permits us to write down master and Fokker-Planck equations that may be simplified under certain, well-defined approximations. These methods allow us to see that memory based on synaptic filters can be viewed as an initial transient that leads to memory signal rise, followed by the emergence of Ornstein-Uhlenbeck-like dynamics that return the system to equilibrium. We may use this approach to compute mean first passage time-defined memory lifetimes for complex models of memory storage.

Variations on the Theme of Synaptic Filtering: A Comparison of Integrate-and-Express Models of Synaptic Plasticity for Memory Lifetimes

Integrate-and-express models of synaptic plasticity propose that synapses integrate plasticity induction signals before expressing synaptic plasticity. By discerning trends in their induction signals, synapses can control destabilizing fluctuations in synaptic strength. In a feedforward perceptron framework with binary-strength synapses for associative memory storage, we have previously shown that such a filter-based model outperforms other, nonintegrative, "cascade"-type models of memory storage in most regions of biologically relevant parameter space. Here, we consider some natural extensions of our earlier filter model, including one specifically tailored to binary-strength synapses and one that demands a fixed, consecutive number of same-type induction signals rather than merely an excess before expressing synaptic plasticity. With these extensions, we show that filter-based models outperform nonintegrative models in all regions of biologically relevant parameter space except for a small sliver in which all models encode memories only weakly. In this sliver, which model is superior depends on the metric used to gauge memory lifetimes (whether a signal-to-noise ratio or a mean first passage time). After comparing and contrasting these various filter models, we discuss the multiple mechanisms and timescales that underlie both synaptic plasticity and memory phenomena and suggest that multiple, different filtering mechanisms may operate at single synapses.

The Enhanced Rise and Delayed Fall of Memory in a Model of Synaptic Integration: Extension to Discrete State Synapses

Integrate-and-express models of synaptic plasticity propose that synapses may act as low-pass filters, integrating synaptic plasticity induction signals in order to discern trends before expressing synaptic plasticity. We have previously shown that synaptic filtering strongly controls destabilizing fluctuations in developmental models. When applied to palimpsest memory systems that learn new memories by forgetting old ones, we have also shown that with binary-strength synapses, integrative synapses lead to an initial memory signal rise before its fall back to equilibrium. Such an initial rise is in dramatic contrast to nonintegrative synapses, in which the memory signal falls monotonically. We now extend our earlier analysis of palimpsest memories with synaptic filters to consider the more general case of discrete state, multilevel synapses. We derive exact results for the memory signal dynamics and then consider various simplifying approximations. We show that multilevel synapses enhance the initial rise in the memory signal and then delay its subsequent fall by inducing a plateau-like region in the memory signal. Such dynamics significantly increase memory lifetimes, defined by a signal-to-noise ratio (SNR). We derive expressions for optimal choices of synaptic parameters (filter size, number of strength states, number of synapses) that maximize SNR memory lifetimes. However, we find that with memory lifetimes defined via mean-first-passage times, such optimality conditions do not exist, suggesting that optimality may be an artifact of SNRs.