Unified co-development of feedforward and intracortical feedback connections in the adapting-synapse model of visual cortical development (Dong and Hopfield, 1992)

Dong and Hopfield (1992) studied recurrent neural networks whose neuronal activities and slowly filtered Hebbian synaptic correlations (synaptic strengths that integrate the co-activity of connected neurons) co-evolve, admitting a Lyapunov function - a scalar, non-increasing energy-like quantity that guarantees convergence to attractors - for symmetric synapses, and used this adapting-synapse framework to show that intracortical feedback connections can self-organize by symmetry breaking into oriented stripe patterns, giving an intrinsic-cortex origin for orientation columnar structure, after which LGN (lateral geniculate nucleus, the thalamic relay of visual input to the cortex) afferent receptive fields become orientation selective with approximately the same spatial period. The paper's discussion leaves open a unified treatment in which feedforward (thalamocortical) and feedback (intracortical) synapses develop simultaneously under realistic conditions: natural visual experience instead of a fixed center-surround input-correlation kernel, binocular inputs, and larger cortical areas supporting multiple interacting columnar systems such as ocular dominance columns. One item of the original discussion (separate excitatory and inhibitory synapse classes with sign constraints, asymmetric connections) is not part of the open core: the authors had already completed the extension to biologically realistic models in the 1991 thesis on which the paper builds. The field-level goal of joint afferent-lateral development was subsequently achieved in a different formalism (self-organizing-map (SOM) based LISSOM models with ocular-dominance demonstrations), and no later work has delivered the unified co-development model or extended the Lyapunov/energy-function convergence analysis to it; this framework-specific core remains open.

Background

Primary visual cortex (V1) is organized into columnar maps: neurons with similar stimulus preferences align vertically through the cortical depth, and preference varies periodically across the surface - most famously for stimulus orientation (orientation columns, first mapped by Hubel and Wiesel in the 1960s) and for the eye of origin (ocular dominance columns). A long-standing question is how such maps self-organize under activity-dependent plasticity, i.e., Hebbian rules under which a synapse strengthens when its pre- and postsynaptic neurons are repeatedly active together.

Models of the 1980s treated this as feedforward learning. Linsker's layered network showed that Hebbian learning through successive processing stages can generate spatial-opponent (center-surround) and orientation-selective receptive fields (the patterns of input activity to which a neuron responds) without a teacher; the BCM theory of Bienenstock, Cooper and Munro introduced input-dependent synaptic modification thresholds; and the correlation-based simulations of Miller, Keller and Stryker reproduced the development of ocular dominance columns from competing thalamocortical afferents. These models evolved synaptic strengths while treating cortical activity as a static function of the input, largely ignoring the real-time recurrent dynamics of cortical circuits - even though V1 contains extensive horizontal (intracortical) connections whose columnar specificity was established by Gilbert and Wiesel.

Dong and Hopfield (1992) analyzed the missing ingredient. Their adapting-synapse network is a recurrent, graded-response Hopfield-type network (each neuron's input is a linear sum of other neurons' activities plus external input, smoothed through a monotone gain function) in which the recurrent synapses are themselves slow dynamical variables that integrate activity correlations:

$$a_i \frac{du_i}{dt} = -u_i + \sum_j T_{ij} V_j + I_i, \qquad V_i = g_i(u_i), \qquad B_{ij} \frac{ds_{ij}}{dt} = -s_{ij} + V_i V_j, \qquad T_{ij} = H_{ij}(s_{ij}),$$

where $u_i$ is the internal state of neuron $i$, $V_i$ its firing rate, $I_i$ external input, $a_i$ a fast activity time constant, $B_{ij} \gg a_i$ a slow synaptic time constant, $s_{ij}$ a low-pass-filtered measure of the correlation of the activities of neurons $i$ and $j$, and $g_i$ and $H_{ij}$ monotone gain functions. For symmetric synapses ($T_{ij} = T_{ji}$) and constant inputs they constructed a Lyapunov function - a scalar quantity that is non-increasing along every trajectory and therefore guarantees convergence to attractors - for the joint activity-synapse dynamics. Learning in this network proceeds in two regimes: while inputs dominate, the synapses average the input correlations (as in an outer-product associative memory, whose weight matrix stores activity patterns through their correlations); once the recurrent connections become strong, the network undergoes symmetry breaking - the dynamics selects and deepens one of several equivalent patterns while suppressing the rest, in the physics sense of a symmetric state losing stability to a less symmetric one.

Applied to a 9x9 patch of cortex driven by inputs with center-surround correlations (mimicking correlated spontaneous activity of the lateral geniculate nucleus, LGN, the thalamic relay of visual input to cortex), the network first develops center-surround intracortical connections; the Fourier spectrum of this connection kernel then favors spatial modes at a characteristic wavelength, and the orientation symmetry breaks into periodic stripes of excitatory and inhibitory intracortical connections. The columnar structure of orientation preference thus originates in the feedback pathway within one cortical area, rather than in feedforward afferents between areas. The 1992 demonstration is deliberately idealized: symmetric synapses, a single neuron class whose synapses can be either excitatory or inhibitory (real cortex obeys Dale's law, under which all output synapses of a neuron carry the same sign), piecewise-linear gains, periodic boundary conditions, a fixed input-correlation structure, monocular input, and a single columnar system.

The authors' own program removed the first idealizations: Dong's 1991 thesis, on which the paper builds, constructs a biologically detailed retina-LGN-cortex model and a developmental variant in which excitatory synapses are constrained to remain nonnegative and inhibitory synapses nonpositive (no sign change, so synapse classes stay distinct) with asymmetric, probabilistically grown connectivity - and shows that the center-surround stage and the orientation symmetry breaking survive. On a 48x48 network the stripes form a locally parallel columnar mosaic with different local orientations, and the afferent (LGN-to-cortex) connections subsequently develop orientation-selective receptive fields whose orientation changes with about the same period as the intracortical stripe pattern; later follow-ups (1993, 1994) recapped this developmental picture and turned to the decorrelation dynamics of the mature cortex. What that program never delivered, and what the 1992 paper's discussion points to as the remaining task, is a unified developmental treatment in which the feedforward and feedback pathways co-develop under realistic visual experience.

Problem Statement

The 1992 paper's discussion, corrected for the biological-realism extension the authors had already completed, leaves open the unification and realism of the adapting-synapse account of visual cortical development. Within the Dong-Hopfield framework (the adapting-synapse class defined by the activity-synapse equations quoted in the Background: a recurrent network whose neuronal activities and slowly filtered Hebbian synaptic correlations co-evolve, admitting a Lyapunov function for symmetric synapses and constant inputs, and whose Dale-constrained variant - nonnegative excitatory and nonpositive inhibitory synapses, asymmetric connectivity - was already shown to preserve the symmetry breaking), construct and analyze a model in which:

(1) the feedforward (LGN-to-cortex, afferent) synapses and the intracortical feedback synapses adapt simultaneously under the same joint activity-synapse dynamics, replacing the sequential scheme in which afferent receptive fields develop only after the intracortical connections have broken orientation symmetry;

(2) the model is driven by realistic visual experience - natural-image input statistics and binocular (two-eye) input - rather than by the fixed center-surround correlation kernel, and it abandons the periodic boundary conditions and the monocular patch of the original demonstrations;

(3) the modeled cortical area is large enough to support multiple interacting columnar systems, in particular orientation preference together with ocular dominance columns. This condition is qualitative rather than a numeric size threshold: it is satisfied when the modeled area supports both an orientation-preference map and ocular-dominance columns as coexisting columnar systems.

For such a unified model, determine the convergence and stability properties of the combined dynamics - specifically whether a Lyapunov/energy-function argument, a contractivity argument, or equivalent guarantees can be extended past symmetric synapses and constant inputs to simultaneous feedforward-intracortical plasticity with sign constraints - and establish whether the characteristic spatial-frequency locking between the afferent orientation map and the intracortical stripe pattern, obtained in the sequential treatment, survives when the two pathways co-develop, that is, whether the relative difference between the two periods is below 10%. The characteristic spatial period of either map is the wavelength of the dominant peak of its radially averaged spatial power spectrum, where the afferent orientation map is the single binocular orientation-preference field rather than one map per eye; the relative difference between the two periods is their absolute difference divided by the larger of the two.

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