Title: The Synaptic Weight Matrix
Description: A key role in simplified models of neural circuitry (Wilson and Cowan, 1972) is played by the matrix of synaptic weights, also called connectivity matrix, whose elements describe the amount of influence the firing of one neuron has on another. Biologically, this matrix evolves over time whether or not sensory inputs are present, and symmetries possessed by the internal dynamics of the network may break up spontaneously, as found in the development of the visual cortex (Hubel and Wiesel, 1977). In this thesis, a full analytical treatment is provided for the simplest case of such a biological phenomenon, a single dendritic arbor driven by correlation-based dynamics (Linsker, 1988). Borrowing methods from the theory of Schrödinger operators, a complete study of the model is performed, analytically describing the break-up of rotational symmetry that leads to the functional specialization of cells. The structure of the eigenfunctions is calculated, lower bounds are derived on the critical value of the correlation length, and explicit expressions are obtained for the long-term profile of receptive fields, i.e. the dependence of neural activity on external inputs.

The emergence of a functional architecture of orientation preferences in the cortex is another crucial feature of visual information processing. This is discussed through a model consisting of large neural layers connected by an infinite number of Hebb-evolved arbors. Ohshiro and Weiliky (2006), in their study of developing ferrets, found correlation profiles of neural activity in contradiction with previous theories of the phenomenon (Miller, 1994; Wimbauer, 1998). The theory proposed herein, based upon the type of correlations they measured, leads to the emergence of three different symmetry classes. The contours of a highly structured phase diagram are traced analytically, and observables concerning the various phases are estimated in every phase by means of perturbative, asymptotic and variational methods. The proper modeling of axonal competition proves to be key to reproducing basic phenomenological features. While these models describe the long-term effect of synaptic plasticity, plasticity itself makes the connectivity matrix highly dependent on particular histories, hence its stochasticity cannot be considered perturbatively.

The problem is tackled by carrying out a detailed treatment of the spectral properties of synaptic-weight matrices with an arbitrary distribution of disorder. Results include a proof of the asymptotic compactness of random spectra, calculations of the shape of supports and of the density profiles, a fresh analysis of the problem of spectral outliers, a study of the link between eigenvalue density and the pseudospectrum of the mean connectivity, and applications of these general results to a variety of biologically relevant examples. The strong non-normality of synaptic-weight matrices (biologically engineered through Dale’s law) is believed to play important functional roles in cortical operations (Murphy and Miller, 2009; Goldman, 2009). Accordingly, a comprehensive study is dedicated to its effect on the transient dynamics of large disordered networks. This is done by adapting standard field-theoretical methods (such as the summation of ladder diagrams) to the non-Hermitian case. Transient amplification of activity can be measured from the average norm squared; this is calculated explicitly for a number of cases, showing that transients are typically amplified by disorder.

Explicit expressions for the power spectrum of response are derived and applied to a number of biologically plausible networks, yielding insights into the interplay between disorder and non-normality. The fluctuations of the covariance of noisy neural activity are also briefly discussed. Recent optogenetic measurements have raised questions on the link between synaptic structure and the response of disordered networks to targeted perturbations. Answering to these developments, formulae are derived that establish the operational regime of networks through their response to specific perturbations, and a minimal threshold is found to exist for counterintuitive responses of an inhibitory-stabilized circuit such as have been reported in Ozeki et al. (2016), Adesnik (2016), Kato et al. (2017). Experimental advances are also bringing to light unsuspected differences between various neuron types, which appear to translate into different roles in network function (Pfeffer et al., 2013; Tremblay et al., 2016).

Accordingly, the last part of the thesis focuses on networks with an arbitrary number of neuronal types, and predictions are provided for the response of networks with a multipopulation structure to targeted input perturbations.