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The Synaptic Weight Matrix The Synaptic Weight Matrix: Dynamics, Symmetry Breaking, and Disorder Francesco Fumarola Submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the Graduate School of Arts and Sciences COLUMBIA UNIVERSITY 2018 © 2018 Francesco Fumarola All rights reserved ABSTRACT The Synaptic Weight Matrix: Dynamics, Symmetry Breaking, and Disorder Francesco Fumarola 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 sponta- neously, 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 biologi- cal 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 calcu- lated, 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 varia- tional 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 stochas- ticity 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 distribu- tion 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 pseu- dospectrum 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 num- ber of neuronal types, and predictions are provided for the response of networks with a multipopulation structure to targeted input perturbations. Contents Acknowledgments xiii 1 Introduction 1 2 Biological foundations 6 2.1 Fundamentals of neural electrodynamics . 6 2.2 Derivation of rate models . 8 2.3 Range of validity . 13 2.4 Equivalence of alternative rate models . 13 3 Hebbian dynamics and the Linsker cell 16 3.1 Hebbian dynamics in the visual cortex . 16 3.1.1 Hebbian dynamics . 16 3.1.2 Visual cortex development . 17 3.1.3 Hebbian theories of orientation selectivity . 20 3.2 Diagonalization of the Linsker operator . 25 3.2.1 Diagonalization in polar coordinates . 25 3.2.2 Diagonalization in Cartesian coordinates . 30 3.2.3 Connecting the polar and Cartesian representations . 32 vi 3.3 Homeostatic constraints . 36 3.3.1 Positive semidefiniteness of the constrained operator . 38 3.3.2 Matrix elements of the constrained operator . 39 3.4 Diagonalization of the constrained model . 41 3.4.1 Long-range solution . 41 3.4.2 Moveable node theory . 42 3.4.3 A lower bound for the transition point . 46 3.5 Conclusions . 49 4 A multilayer model of orientation development 52 4.1 The Hebbian theory of orientation maps . 52 4.1.1 Review of the experimental literature . 52 4.1.2 Unconstrained dynamics of the model . 56 4.1.3 Homeostatic constraint . 60 4.1.4 Projection operators . 63 4.2 Properties of the time-evolution operator . 64 4.2.1 Matrix elements . 65 4.2.2 Fourier Transform . 66 4.2.3 Positive semidefiniteness . 68 4.2.4 Long-term dynamics . 69 4.2.5 Symmetries of the system: translations and rotations . 70 4.2.6 Symmetries of the system: parity and CP symmetry . 71 4.2.7 Diagonalization of the unconstrained dynamics . 74 4.3 Perturbation theory for short correlations . 76 4.3.1 Solution with uncorrelated input . 76 4.3.2 Optimal wavenumber for oriented eigenfunctions . 79 4.3.3 Optimal wavenumber for non-oriented eigenfunctions . 81 4.4 Forbidden regions for the N- and R-phases . 84 4.5 The long-range limit . 86 4.5.1 The long-range limit: derivation . 86 4.5.2 The long-range limit: analysis of results . 93 4.6 Cartesian-basis expansion . 96 4.6.1 Combining s-waves with longitudinal eigenfunctions (0x theory) . 99 4.6.2 Short-range limit . 101 4.6.3 Local stability of the R-phase . 104 4.6.4 0x eigenfunction . 106 4.6.5 The uniform phase . 106 4.7 The N-phase domain . 108 4.7.1 Global stability of the 0x solution . 108 4.7.2 Shape of the phase boundary . 109 4.7.3 Corrections to the variational results . 112 4.8 Numerics and conclusions . 120 4.8.1 Numerical results . 120 4.8.2 Conclusions . 121 5 The Spectral Properties of Weight Matrices 125 5.1 The open problem of synaptic stochasticity . 125 5.1.1 Background and motivation . 125 5.1.2 Random matrices beyond quantum mechanics . 126 5.1.3 Outline of the treatment . 129 5.2 General results on spectral properties . 130 5.2.1 Preliminary definitions . 130 5.2.2 The shape of supports and the density profiles . 133 5.2.3 Formulation in terms of singular values . 135 5.2.4 Purely random synaptic weight matrices . 136 5.2.5 Ordering of limits . 137 5.2.6 Relationship to pseudospectra . 141 5.3 Calculation of spectral properties . 144 5.3.1 The shape of supports and density profiles . 144 5.3.2 Singular value formulation . 157 5.3.3 Validity of the non-crossing approximation . 158 5.3.4 Proving the compactness of the supports . 169 5.3.5 The case of zero mean connectivity . 175 5.4 Applications to specific networks . 180 5.4.1 Schur structure of the mean connectivity . 181 5.4.2 Networks with a single feedforward chain . 182 5.4.3 Examples motivated by Dale’s law . 185 5.4.4 Linearizations of nonlinear networks . 201 5.5 Conclusions . 206 6 Disordered neural dynamics 208 6.1 Dynamical functions of non-normality . 208 6.2 General results . 211 6.2.1 Preliminary remarks . 211 6.2.2 Result for the average squared norm . 212 6.2.3 Results for the power spectrum . 213 6.2.4 The amplifying effects of disorder . 216 6.2.5 Conditions for applicability . 217 6.3 Derivation of the general results . 219 6.3.1 Formulation in terms of diffusive kernel . 219 6.3.2 Summing up the diffuson . 224 6.4 Applications to specific networks . 232 6.4.1 Feedforward chain with alternating eigenvalues . 233 6.4.2 Feedforward chain with null eigenvalues . 235 6.4.3 Network with two-state feedforward components . 239 6.5 Beyond the non-crossing approximation . 242 6.6 Disorder fluctuations of the covariance . 244 7 Response to targeted input perturbations 249 7.1 Measuring the response to input perturbations . 249 7.2 Modeling paradoxical responses . 252 7.2.1 Are excitatory subnetworks independently stable? . 252 7.2.2 A uniform excitatory-inhibitory network .
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