Blais, Cooper and Shouval (Neural Computation, 2000) showed in a feedforward model with two temporally offset LGN input channels (lagged and non-lagged cells) trained on natural scenes that a second-order PCA-like Hebbian rule cannot develop direction-selective (DS) receptive fields while BCM-like higher-order rules can, that DS forms only within an intermediate window of drift velocity (partial temporal overlap of the two channels), and that the no-overlap regime reproduces the loss of DS with preserved orientation selectivity caused by strobe rearing. The work itself left the biological robustness of these conclusions unresolved. The open problem is to determine, quantitatively and in natural-scene environments, whether the conclusions survive biologically realistic generalizations: (1) replacing the two discrete delay channels by multiple graded delay channels or continuous time-dependent synaptic weights, as in ICA formulations of spatiotemporal receptive fields; and (2) adding recurrent cortico-cortical interactions, establishing whether and to what extent they quantitatively modify DS formation and under which conditions they become required for it. These facets jointly test whether the single-cell two-channel plasticity-rule account of DS formation is an artifact of its simplifications.

Background

Most simple cells in the primary visual cortex (V1) of cats, ferrets and primates are tuned not only to the orientation of visual edges but also to the direction of stimulus motion: at the preferred orientation, the cell responds more strongly to a drifting grating moving one way than to the identical grating moving the opposite way. In a linear approximation the cell's response is the convolution of the visual input with a spatiotemporal receptive-field kernel K(x,t)K(x,t), and a direction-selective cell must have a space-time inseparable kernel, one that cannot be factorized into a purely spatial and a purely temporal factor, because motion detection requires comparing the stimulus at different spatial positions at different times. A natural feedforward source of such differential timing is the lateral geniculate nucleus (LGN), the thalamic relay between retina and cortex, which contains lagged and non-lagged cells that sample the same visual space but differ in response timing: in carnivores, lagged cells relay the same retinal signal substantially later than non-lagged cells, so the two channels provide temporally offset copies of the visual input.

Whether DS receptive fields can self-organize under biologically plausible synaptic plasticity has been studied since the 1990s. Feidler, Saul, Murthy and Humphrey (1997) showed, in a model neuron driven by a few sinusoidally varying inputs, that simple Hebbian rules driven only by input covariance are inadequate for DS development, whereas rules with the BCM property succeed: in the Bienenstock-Cooper-Munro (BCM) rule (Bienenstock, Cooper and Munro, 1982), synaptic modification is negative for small postsynaptic activity and positive for large postsynaptic activity, with a modification threshold that slides as a running average of recent activity, so that the rule is sensitive to higher-than-second-order statistics. Wimbauer, Wenisch, Miller and van Hemmen (1997, parts I and II) formulated correlation-based models with continuous LGN temporal filters and cortical network interactions, finding that DS develops only if lagged and non-lagged inputs are nearly uncorrelated or the environment contains anisotropic motion. In parallel, independent component analysis (ICA), an unsupervised method that linearly transforms inputs so that the outputs are as statistically independent as possible, was applied by van Hateren and Ruderman (1998) to natural image sequences; it yielded space-time inseparable, simple-cell-like spatiotemporal filters, using many discrete temporal delays and requiring spatiotemporal whitening (decorrelation) of the input.

Blais, Cooper and Shouval (2000) asked which of these conclusions survive a natural-scene environment, introduced as a training paradigm for BCM-type rules by Law and Cooper (1994). A single model cortical cell received two spatially coextensive LGN input channels, one lagged and one non-lagged, and was trained on 13-by-13-pixel patches from twelve natural-scene images, spatially preprocessed by a difference-of-Gaussians filter emulating retinal processing, that moved across the receptive field in a sequence of random saccades and constant-velocity drifts; the LGN lag was fixed at one simulation iteration. Four single-cell plasticity rules were compared: Oja's stabilized Hebbian rule (Oja, 1982), which extracts the first principal component of the input (a PCA, second-order-statistics rule); the quadratic BCM rule; and two BCM-like rules based on the statistical moments of skewness and kurtosis (Blais, Intrator, Shouval and Cooper, 1998). The findings were: first, a linear-algebraic result shows that for a spatially isotropic two-channel input whose covariance has the block-symmetric form

C=(C1C2C2C1),\mathbf{C}=\begin{pmatrix}\mathbf{C}_1 & \mathbf{C}_2 \\ \mathbf{C}_2 & \mathbf{C}_1\end{pmatrix},

every principal-component eigenvector v=(v1,v2)\mathbf{v}=(\mathbf{v}_1,\mathbf{v}_2) has channel profiles that are identical up to sign, v1=±v2\mathbf{v}_1=\pm\mathbf{v}_2, so the PCA rule necessarily produces space-time separable, non-DS receptive fields; second, the BCM, skewness and kurtosis rules all develop temporally shifted lagged and non-lagged profiles, that is, space-time inseparable and direction-selective receptive fields; third, with the lag fixed, DS develops only for intermediate drift velocities (partial temporal overlap of the two channels' input patterns) and is lost both at very low velocity (the two channels become identical) and at very high velocity (no temporal overlap; only one channel survives on the cortical cell). The high-velocity regime was identified with strobe rearing, rearing animals under stroboscopic illumination, which experimentally abolishes cortical DS while sparing orientation selectivity (Cynader and Chernenko, 1976; Humphrey and Saul, 1998) and prevents the convergence of inputs with different response timings onto single cortical cells (Humphrey, Saul and Feidler, 1998) - phenomenology the simulations reproduce. Fourth, training on natural movies with more realistic LGN temporal filters left the qualitative conclusions unchanged. DS was measured by the standard direction-selectivity index computed from responses to drifting gratings at the preferred orientation and spatial frequency.

The paper closes by acknowledging what remains unresolved. It treats a purely feedforward, single-cell, two-channel abstraction, while noting that cortical-cortical interaction likely also plays a part and describing the work as a first step towards a network model. Its temporal parameters (the one-iteration LGN lag, the specific temporal-filter shapes, drift velocities in pixels per iteration) are simulation units with no quantitative mapping to measured latencies, filter time courses or eye-movement velocities in real animals, so it is unknown which in-vivo temporal regimes correspond to the DS-permitting velocity window. The claim that the studied rules neither require whitening nor are adversely affected by it is qualitative: no systematic characterization of how the degree of input whitening or the amplitude of residual correlations affects DS formation across rules is given, even though ICA-type rules are sensitive to small residual correlations in nearly whitened environments. Finally, the relation to the multi-delay ICA picture is left open: the ICA results can be read either as many LGN populations with different time lags converging through independent synaptic weights (a graded-delay generalization of the two-channel model) or as single synapses carrying weights that differ at different time points, and whether the qualitative conclusions about which learning rules produce DS survive such multi-delay or continuous-time-weight architectures, with or without network interactions, was not determined.

Problem Statement

Determine, quantitatively and within natural-scene environments, whether the central conclusions of the two-channel single-cell account of direction-selectivity (DS) formation - (a) second-order (PCA-like) Hebbian rules fail while BCM-like higher-order rules succeed, and (b) DS forms only within an intermediate temporal-overlap (drift-velocity) window - survive biologically realistic generalizations, and establish which plasticity mechanisms can account for DS development under realistic input statistics and circuit architecture. Within the lagged/non-lagged geniculocortical framework trained on natural-scene (image or movie) inputs, the open problem has two facets:

  1. Multi-delay architectures: replace the two discrete delay channels by multiple graded delay channels or continuous time-dependent synaptic weights, as in ICA formulations of spatiotemporal receptive fields, and determine whether and how this changes which learning rules produce DS and the drift-velocity window over which DS forms. All of the following sub-questions are required: (i) whether conclusion (a), the PCA-fails/higher-order-rules-succeed dichotomy, persists for graded-delay inputs; (ii) whether its covariance block-symmetry explanation accounts for the outcome; and (iii) whether conclusion (b), the intermediate temporal-overlap (drift-velocity) window, persists for graded-delay inputs.

  2. Cortical recurrence: extend the model with recurrent cortico-cortical interactions (which may themselves be plastic) and determine whether and to what extent they quantitatively modify DS formation relative to the single-cell feedforward mechanism, including the input or architecture conditions under which recurrent interactions become required for DS formation (for example graded-delay inputs on which the single-cell rule fails).

These two requests are facets of a single scientific objective, testing the biological robustness of the single-cell two-channel plasticity-rule account of DS formation. To remain comparable with the source framework, DS should be measured with the standard direction-selectivity index computed from responses to drifting gratings at the preferred orientation and spatial frequency, and the compared rules should span at least one second-order (PCA-type) rule and higher-order (BCM-type) rules.

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