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

Biological Cybernetics (2024) 118:39–81 https://doi.org/10.1007/s00422-024-00985-0 ORIGINAL ARTICLE Stability against fluctuations: a two-dimensional study of scaling, bifurcations and spontaneous symmetry breaking in stochastic models of synaptic plasticity Terry Elliott1 Received: 12 September 2022 / Accepted: 12 February 2024 / Published online: 7 April 2024 © The Author(s) 2024 Abstract 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. Keywords Synaptic plasticity · Neuronal development · Stochastic processes · Fluctuations 1 Introduction Spike-timing-dependent plasticity (STDP; Markram et al. 1997; Bi and Poo 1998; Zhang et al. 1998; Froemke and Dan 2002; Roberts and Bell 2002; Harvey and Svoboda 2007; Caporale and Dan 2008)—the biphasic, graded change in synaptic efficacy depending on the relative timing of pre- and postsynaptic spiking—is typically understood to imply that single synapses can express finely graded changes in synaptic strength, with this assumption implicit in many models (Song et al. 2000; van Rossum et al. 2000; Castellani et al. 2001; Senn et al. 2001; Sjöström and Nelson Communicated by Anthony Burkitt. B 1 Terry Elliott Department of Electronics and Computer Science, University of Southampton, Highfield, Southampton SO17 1BJ, UK 2002; Burkitt et al. 2004; Bi and Rubin 2005; Rubin et al. 2005; Bender et al. 2006). Yet some experimental evidence suggests that synapses may occupy only discrete states of synaptic strength (Montgomery and Madison 2002, 2004; O’Connor et al. 2005a, b; Bartol et al. 2015) or may change their strengths only in discrete, all-or-none jumps (Petersen et al. 1998; Yasuda et al. 2003; Bagal et al. 2005; Sobczyk and Svoboda 2007). One way to resolve this apparent contradiction is to propose that a single synapse may express only fixed-amplitude steps in synaptic strength, with the probability but not amplitude of change depending on spike timing (Appleby and Elliott 2005). The classic, graded biphasic STDP curve (Bi and Poo 1998) then emerges as an average change over multiple synapses for a single spike-pair presentation or at a single synapse over multiple spike-pair presentations (Appleby and Elliott 2005). Any probabilistic or stochastic model of synaptic plasticity faces the challenge posed by destabilising fluctuations in synaptic strength. When considering, for example, neu- 123 40 ronal development (Purves and Lichtman 1985), fluctuations can lead to a change in the patterns of synaptic connectivity acquired through activity-dependent competitive dynamics in the developing primary visual cortex (V1). In particular, the segregated afferent input to V1 neurons, in which one eye or the other dominates in the control of V1 neurons (Hubel and Wiesel 1962), can be destabilised by fluctuations, so that the two afferents repeatedly switch control of a target cell over some characteristic, average time scale (Appleby and Elliott 2006; Elliott 2008). Reducing the size of the all-ornone steps in synaptic strength can control these fluctuations, but if the plasticity step size must be very small to control fluctuations, then models become biologically implausible and, furthermore, synaptic strengths become for all practical purposes graded rather than discrete (Elliott 2008). We have instead proposed that synapses act as low-pass filters, integrating their plasticity induction signals before expressing a step change in synaptic strength (Elliott 2008; Elliott and Lagogiannis 2009). These “integrate-and-express” models powerfully control fluctuations without having to resort to implausible parameter choices or assumptions. Previously we extensively analysed three such models of synaptic filtering operating in concert with our model of STDP (Elliott 2011b). We found that the plasticity step size plays the role of a temperature-like parameter, with smaller step sizes corresponding to lower temperatures. Structure formation (segregated states) emerges only below a critical plasticity step size, akin to a Curie point or critical temperature. Synaptic filtering “cools” the dynamics, with the filter threshold scaling back the actual plasticity step size into an effective step size, where this scaling is linear over a range of parameters. This analysis was performed using the master equation for the strengths of two afferents (representing inputs via the lateral geniculate nucleus from the two eyes) and its Fokker–Planck equation limit. In order to write down the master equation, it was necessary to make several approximations, but the key one for the purposes of analytical tractability was to assume that the occurrence of a plasticity step in one afferent resets the filter states in all afferents. This resetting or renewal approximation essentially reduces a random walk in multiple dimensions (afferents) to a set of one-dimensional random walks, one for each afferent. The afferents’ dynamics are then coupled only through the common postsynaptic firing rate of their target cell. We argued although did not demonstrate that this approximation works because our stochastic model of STDP, together with synaptic filtering, decorrelates synaptic strength changes across multiple afferents, with simultaneous changes becoming increasing (...truncated)