Analysis of Activity Dependent Development of Topographic Maps in Neural Field Theory with Short Time Scale Dependent Plasticity

TL;DR

This work develops a continuous neural-field framework to study how complex spatio-temporal activity refines topographic maps. By coupling an activity-driven feed-forward map with a static recurrent network and a plasticity window, the authors derive analytic conditions for stable, refined retinotopy and validate predictions against wild-type and $\beta2$ knockout mouse data using MCMC parameter estimation. A key finding is that the time scale of the plasticity window around $0.56$ s, together with wave-speed and wave-width, shapes the final arborisation, offering an explanatory link to broader retinotopic development and mutant phenotypes. The model demonstrates that biological noise can stabilize development and provides testable predictions for plasticity dynamics, with code available for reproduction.

Abstract

Topographic maps are a brain structure connecting pre-synpatic and post-synaptic brain regions. Topographic development is dependent on Hebbian-based plasticity mechanisms working in conjunction with spontaneous patterns of neural activity generated in the pre-synaptic regions. Studies performed in mouse have shown that these spontaneous patterns can exhibit complex spatial-temporal structures which existing models cannot incorporate. Neural field theories are appropriate modelling paradigms for topographic systems due to the dense nature of the connections between regions and can be augmented with a plasticity rule general enough to capture complex time-varying structures. We propose a theoretical framework for studying the development of topography in the context of complex spatial-temporal activity fed-forward from the pre-synaptic to post-synaptic regions. Analysis of the model leads to an analytic solution corroborating the conclusion that activity can drive the refinement of topographic projections. The analysis also suggests that biological noise is used in the development of topography to stabilise the dynamics. MCMC simulations are used to analyse and understand the differences in topographic refinement between wild-type and the $\beta2$ knock-out mutant in mice. The time scale of the synaptic plasticity window is estimated as $0.56$ seconds in this context with a model fit of $R^2 = 0.81$.

Figures (6)

• Figure 1: The connections and directionality of the model: activity is feed-forward from the pre-synaptic region by the structure of interest and is spatial-temporally propagated by a time-differential operator and spatial convolution of inhibitory and excitatory recurrent connections. We show cartoons of typical propagating activity patterns and recurrent connections but not feed-forward connections as determining these are the object of this study. The generated signal in the post-synaptic region and the driving signal in the pre-synaptic region are then convolved with a plasticity window to inform the synaptic changes on a slow time scale which is indicated by the variable $T = \epsilon t$ for some small $\epsilon$.
• Figure 2: Two examples of topographic organisation using a Wizard hat style function: a) shows a linear relationship between axes and for while b) shows a cubic relationship between axes. Both are topographic but a) has an even representation of the pre-synaptic field across the post-synaptic field while b) compresses the representation at the boundary and enlarges the interior.
• Figure 3: The manifold in $(c,\sigma_2)$ space which defines the stability of the final organisation. Below the surface solutions do not exhibit singularities and the training function is deemed to be stable; in general, small choices for the parameters exhibit stable synaptic organisations at the cost of arbitrarily small amplitude. The manifold appears to be well-above reasonable estimates for these parameters, ensuring the model is likely stable in plausible biological scenarios.
• Figure 4: A typical organisation generated with the parameters shown in Table \ref{['table:parameters']} with (a) showing the representation in Fourier space, and (b) the representation in real space after re-normalisation.
• Figure 5: The variation in width ($\Omega$) in four distinct planar slices of the manifold of parameters which influence the models prediction of mean distribution width. Panel (a) shows that width decreases both with wave-speed and wave-width, qualitatively accounting for the differences between the wild-type and $\beta2$ mutant. Panel (b) shows that width decreases with with the ratio of excitation to inhibition in the recurrent connections $W$ suggesting a smaller zone of excitatory support decreases arbor size. There is an anti-symmetry along the line $r_1 = r_2$ which is expected as the dominant connection type switches along this line. Panel (c) shows that width decreases with recurrent connection amplitude but the effect is not substantial. Panel (d) shows that width predominately decreases in accordance with the plasticity window time-scale, and while the activity time-scale has an effect it is not substantial.

Theorems & Definitions (2)

• Lemma 4.1
• proof