Stability against fluctuations: a two-dimensional study of scaling, bifurcations and spontaneous symmetry breaking in stochastic models of synaptic plasticity

Stochastic models of synaptic plasticity must confront the corrosive influence of fluctuations in synaptic strength on patterns of synaptic connectivity. To solve this problem, we have proposed that synapses act as filters, integrating plasticity induction signals and expressing changes in synaptic strength only upon reaching filter threshold. Our earlier analytical study calculated the lifetimes of quasi-stable patterns of synaptic connectivity with synaptic filtering. We showed that the plasticity step size in a stochastic model of spike-timing-dependent plasticity (STDP) acts as a temperature-like parameter, exhibiting a critical value below which neuronal structure formation occurs. The filter threshold scales this temperature-like parameter downwards, cooling the dynamics and enhancing stability. A key step in this calculation was a resetting approximation, essentially reducing the dynamics to one-dimensional processes. Here, we revisit our earlier study to examine this resetting approximation, with the aim of understanding in detail why it works so well by comparing it, and a simpler approximation, to the system's full dynamics consisting of various embedded two-dimensional processes without resetting. Comparing the full system to the simpler approximation, to our original resetting approximation, and to a one-afferent system, we show that their equilibrium distributions of synaptic strengths and critical plasticity step sizes are all qualitatively similar, and increasingly quantitatively similar as the filter threshold increases. This increasing similarity is due to the decorrelation in changes in synaptic strength between different afferents caused by our STDP model, and the amplification of this decorrelation with larger synaptic filters.

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.

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.

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

We study memory lifetimes in a perceptron-based framework with binary synapses, using the mean first passage time for the perceptron's total input to fall below firing threshold to define memory lifetimes. Working with the simplest memory-related model of synaptic plasticity, we may obtain exact results for memory lifetimes or, working in the continuum limit, good analytical approximations that afford either much qualitative insight or extremely good quantitative agreement. In one particular limit, we find that memory dynamics reduce to the well-understood Ornstein-Uhlenbeck process. We show that asymptotically, the lifetimes of memories grow logarithmically in the number of synapses when the perceptron's firing threshold is zero, reproducing standard results from signal-to-noise ratio analyses. However, this is only an asymptotically valid result, and we show that extending its application outside the range of its validity leads to a massive overestimate of the minimum number of synapses required for successful memory encoding. In the case that the perceptron's firing threshold is positive, we find the remarkable result that memory lifetimes are strictly bounded from above. Asymptotically, the dependence of memory lifetimes on the number of synapses drops out entirely, and this asymptotic result provides a strict upper bound on memory lifetimes away from this asymptotic regime. The classic logarithmic growth of memory lifetimes in the simplest, palimpsest memories is therefore untypical and nongeneric: memory lifetimes are typically strictly bounded from above.

The Rise and Fall of Memory in a Model of Synaptic Integration

Plasticity-inducing stimuli must typically be presented many times before synaptic plasticity is expressed, perhaps because induction signals gradually accumulate before overt strength changes occur. We consider memory dynamics in a mathematical model with synapses that integrate plasticity induction signals before expressing plasticity. We find that the memory trace initially rises before reaching a maximum and then falling. The memory signal dissociates into separate oblivescence and reminiscence components, with reminiscence initially dominating recall. In radical contrast, related but nonintegrative models exhibit only a highly problematic oblivescence. Synaptic integration mechanisms possess natural timescales, depending on the statistics of the induction signals. Together with neuromodulation, these timescales may therefore also begin to provide a natural account of the well-known spacing effect in the transition to late-phase plasticity. Finally, we propose experiments that could distinguish between integrative and nonintegrative synapses. Such experiments should further elucidate the synaptic signal processing mechanisms postulated by our model.

Stability against fluctuations: scaling, bifurcations, and spontaneous symmetry breaking in stochastic models of synaptic plasticity

In stochastic models of synaptic plasticity based on a random walk, the control of fluctuations is imperative. We have argued that synapses could act as low-pass filters, filtering plasticity induction steps before expressing a step change in synaptic strength. Earlier work showed, in simulation, that such a synaptic filter tames fluctuations very well, leading to patterns of synaptic connectivity that are stable for long periods of time. Here, we approach this problem analytically. We explicitly calculate the lifetime of meta-stable states of synaptic connectivity using a Fokker-Planck formalism in order to understand the dependence of this lifetime on both the plasticity step size and the filtering mechanism. We find that our analytical results agree very well with simulation results, despite having to make two approximations. Our analysis reveals, however, a deeper significance to the filtering mechanism and the plasticity step size. We show that a filter scales the step size into a smaller, effective step size. This scaling suggests that the step size may itself play the role of a temperature parameter, so that a filter cools the dynamics, thereby reducing the influence of fluctuations. Using the master equation, we explicitly demonstrate a bifurcation at a critical step size, confirming this interpretation. At this critical point, spontaneous symmetry breaking occurs in the class of stochastic models of synaptic plasticity that we consider.