Mechanisms for Spontaneous Symmetry Breaking in Developing Visual Cortex

For the brain to recognize local orientations within images, neurons must spontaneously break the translation and rotation symmetry of their response functions—an archetypal example of unsupervised learning. The dominant framework for unsupervised learning in biology is Hebb’s principle, but how Hebbian learning could break such symmetries is a longstanding biophysical riddle. Theoretical studies argue that this requires inputs to the visual cortex to invert the relative magnitude of their correlations at long distances. Empirical measurements have searched in vain for such an inversion and report the opposite to be true. We formally approach the question through the Hermitianization of a multilayer model, which maps it into a problem of zero-temperature phase transitions. In the emerging phase diagram, both symmetries break spontaneously as long as (i) recurrent interactions are sufficiently long range and (ii) Hebbian competition is duly accounted for. A key ingredient for symmetry breaking is competition among connections sprouting from the same afferent cell. Such a competition, along with simple monotonic falloff of input correlations with distance, is capable of triggering the broken-symmetry phase required by image processing. We provide analytic predictions on the relative magnitudes of the relevant length scales needed for this novel mechanism to occur. These results reconcile experimental observations to the Hebbian paradigm, shed light on a new mechanism for visual cortex development, and contribute to our growing understanding of the relationship between learning and symmetry breaking.

Popular Summary

The images we see every day are full of light and dark edges of various orientations. It is no surprise that a baby animal learns rapidly to recognize them. Each cell in the brain’s primary visual cortex learns to send electrical signals only when it detects a light or dark edge of a particular orientation. What baffled early experimenters is that this seemed to happen before babies first opened their eyes. How on Earth could babies learn to see without seeing? Here, we provide a theory for this development that finally jibes with observed patterns of neuronal firing.

A long-standing idea is that this ability to detect a specifically oriented edge develops through learning rules by which “neurons that fire together, wire together.” That is, if the firing of one cell helps excite another cell to fire, the connection between them strengthens. This leads to the development of orientation selectivity if the spontaneous firing (generated by the brain in the absence of vision) of the inputs to the visual cortex is correlated in a certain pattern. However, experiments 15 years ago failed to find that pattern, and no theory has been proposed given the patterns that have been observed.

The new theory uses the fact that the inputs that connect more strongly to the cortex compete less effectively for further connections, and vice versa. In the end, all inputs have a similar strength of connection to the cortex. It turns out that this mechanism and the observed correlations suffice to explain orientation-selectivity development.

Having a theory for orientation-selectivity development, scientists can now address the development of further aspects of the visual system such as the variation of preferred orientations in a beautiful periodic pattern across the cortex.

Figure 1

Schematic depiction of the model. Primary visual cortex (V1) is the first receiving area in the cerebral cortex for visual sensory information. Here it is represented by a single postsynaptic layer of cells depicted by the upper row of larger circles. Inputs to V1 come from the lateral geniculate nucleus of the thalamus (LGN), which in turns takes input from the eyes. LGN cells are excited either by light onset or by light offset on the corresponding spot of the retina, and these two types of cells are represented by the two rows of smaller circles. The quantity of synapses from a LGN location $\mathit{\alpha }$ to a V1 location $\mathit{z}$ is described by an “arbor density” $A(\mathit{z}-\mathit{\alpha })$. Their total synaptic strength is defined as either of the functions $s^{\mathrm{ON}}(\mathit{z},\mathit{\alpha })$ or $s^{\mathrm{OFF}}(\mathit{z},\mathit{\alpha })$ depending on whether the presynaptic cell is of the ON or OFF type. Correlations between the activities of two presynaptic cells of types $(i)\in \{\text{ON},\text{OFF}\}$ and $(j)\in \{\text{ON},\text{OFF}\}$ located at retinotopic positions $\mathit{\alpha }$ and $\mathit{\beta }$ are described by a set of functions $C^{(i,j)}(\mathit{\alpha }-\mathit{\beta })$. The effect via lateral connections (brown arrows) of activity at cortical position $\mathit{x}$ on activity at cortical position $\mathit{y}$ is characterized by the function $I(\mathit{x}-\mathit{y})$.

Figure 2

Phase diagram of the model. (a) Structure of the phase diagram resulting from analytical investigation of the time evolution. The model’s parameters are the correlation length $\zeta$ of the inputs from the LGN, the radius of outgoing connections $\rho$, and the length scale $\eta$ of lateral connections in V1. A phase diagram may use for coordinates any pair of dimensionless ratios among these parameters; here, $\zeta /\rho$ and $\eta /\rho$ are used. The labels $N$, $R$, $T$ correspond, respectively, to no symmetry breaking, breaking of rotational symmetry only, and breaking of both translational and rotational symmetry (cf. Table 1). The red dot marks the inferred location of a triple point. (b) Dominant wave number of map modulation across the cortex in units of the inverse arbor radius. The preservation of cortically uniform states is confirmed in the regions corresponding to the $R$ and $N$ phases. Clearly visible are the sharp boundaries of the two cortically uniform phases $N$ and $R$ separated by the nonuniform phase $T$, where orientation maps arise. The plots are obtained by diagonalizing a discretized version of the operator $L^p$ defined by Eq. (a36), where RFs are confined to a square of side equal to six arbor radii represented by a $15\times 15$ grid. (c) Larger-scale view of the phase diagram displaying as a gray dotted line the asymptotic phase boundary ${\eta }_c\sim \sqrt{2}\zeta$ obtained from an analytic solution of the model.

Figure 3

Receptive fields by symmetry class. Plots of the receptive fields of Eqs. (1), (2), and (3), respectively, for the $N$, $R$, and $T$ phases. Hue scale ranges between minimum and maximum values, allowing for an arbitrary scaling factor. Functions are evaluated at representative points $P=(\zeta /\rho ,\eta /\rho )$ in parameter space given by $P_N=(0.02,0.2)$, $P_R=(0.05,0.7)$, $P_T=(5,3)$ (see Appendix pp3 for numerical results with the same parameters). The relative length of an arbor radius $\rho$ for all plots is shown to the left of the figure. The receptive field for the $R$ phase is plotted with a randomly chosen orientation. For the $T$ phase, the function $s_T$ of Eq. (3) is rotated by the complex angle ${\varphi }_0=\arctan (-\int \Im s_T/\int \Re s_T)$ so as to make the imaginary part odd under inversion of the cortical modulation axis (the real part becomes symmetric as a consequence; see Appendix Sec. pp2-s2), and we separately display its real (left) and imaginary (right) components. The fastest-growing mode of time evolution is obtained by multiplying receptive field by a modulating phase factor $e^{i\omega x}$, with $x$ being the coordinate for the degenerate direction of modulation in cortical space.

Figure 4

Results of dynamical simulations. Simulations of the equations of motion [Eqs. (a7)] are performed on discretized receptive fields evolving from random initial conditions while waiting for the configurations to stabilize. We use $32\times 32$ grids of cortical cells and of ON and OFF LGN cells, with periodic boundary conditions. Each LGN cell projects an arbor of inputs to the cortex that is nonzero over a circle of diameter 13 centered on its corresponding point in the cortical grid. (a) Large-time configuration of $s^{\mathrm{ON}}$ after evolving the constraint coming from Hebbian competition. Each $13\times 13$ square contains the diameter-13 (or smaller) set of ON weights to one cortical cell. A $32\times 32$ grid of cortical cells is illustrated. (b) As in panel (a) but for $s^{\mathrm{OFF}}$, which yields analogous breaking of the symmetries. (c) Difference between $s^{\mathrm{ON}}$ and $s^{\mathrm{OFF}}$ of panels (a) and (b) with a portion expanded to highlight cell-by-cell structure [panel (d)]. (e) Large-time state of $s^{\mathrm{ON}}$ evolved without the competitive constraint [Eq. (a6)] and displaying indeed no trace of either symmetry breaking. In this fully symmetric solution, all RFs are identical and lacking orientation tuning. Parameters used for all these simulations: $\eta /\rho =0.75,\zeta /\rho =0.25,\rho =6.5$.

Figure 5

Invariances of the theory. Depiction of the two main transformations under which the model is invariant: simultaneous rotation and simultaneous translation of the three neuron layers. Notice that the receptive field coordinate $\mathit{r}$ is left untransformed by translations.

Figure 6

Receptive fields for the $N$ phase. Example of the variational RF of Eq. (1) in the main text compared to the result of numerically diagonalizing the full operator ${\widehat{L}}^p$ and rescaling the eigenfunction by Eq. (a32). The parameters used here are $\zeta /\rho =0.02$, $\eta \rho =0.2$. A side of the grid has length equal to $5\rho$; color scale ranges between min and values.

Figure 7

Variational landscape for cortically uniform phases. Expectation values of the constrained time-evolution operator ${\widehat{L}}^f$ (in units of $2\pi {\mu }^2$) plotted as a function of $R/\rho$. The three curves refer to (i) ${\lambda }_{0,1}$ (expectation value of ${\widehat{L}}^{\mu }$ or ${\widehat{L}}^f$ in the exact eigenfunction ${\mathrm{\Psi }}_{0,1}$) plotted in green; (ii) ${\mathcal{E}}_0$, expectation value of the unconstrained operator ${\widehat{L}}^{\mu }$ in the moveable-node state ${\psi }_T$ plotted in blue; (iii) $\mathcal{E}$, expectation value of the constrained operator ${\widehat{L}}^f$ in the state ${\psi }_T$ plotted in purple. The figure refers to $h=(\mu /\rho )=.01$. Values of $R$ for which the moveable-node state is preferred to the orientation-selective state are different for the two operators ${\widehat{L}}^{\mu }$ and ${\widehat{L}}^f$. Namely, there exists a minimal value $\tilde{R}$, in this example being approximately equal to 0.14, such that ${\widehat{L}}^{\mu }$ opts for ${\psi }_T$ at sufficiently high values of the node radius $R>\tilde{R}$, while ${\widehat{L}}^f$ does so for values of $R$ in a narrow window $R\gtrsim \tilde{R}$.

Figure 8

Structure of cortically uniform phases over the phase diagram. Insets show plots of the principal eigenfunctions, respectively, the movable-node approximation for the $N$ phase and the exact eigenfunctions with longitudinal and transverse alignments for the $R$ phase.

Figure 9

Receptive fields for the $R$ phase. Example of the RF of Eq. (2) compared to the result of numerically diagonalizing the full operator ${\widehat{L}}^p$ and rescaling the eigenfunction by Eq. (a32). The parameters used here are $\zeta /\rho =0.05$, $\eta \rho =0.7$. A side of the grid has length equal to $5\rho$; color scale ranges between min and values.

Figure 10

Receptive fields for the $T$ phase. Plots of the symmetric and antisymmetric components of Eq. (3) in the main text to the Fourier-transformed eigenfunction compared to the result of numerically diagonalizing the full operator ${\widehat{L}}^p$ and rescaling the principal eigenfunction by Eq. (a32). Receptive fields $\tilde{s}$ are rotated by the complex angle ${\varphi }_0=\arctan (-\int \Im \tilde{s}/\int \Re \tilde{s})$ so as to make the imaginary part odd under inversion of the cortical modulation axis (the real part becomes symmetric as a consequence; see Sec. pp2-s2). The parameters used here are $\zeta =5\rho$ and $\eta =3\rho$ corresponding to a ground state at $\omega \approx 0.48/\rho$. A side of the grid has length equal to $5\rho$; color scale ranges between min and values.

Figure 11

Patching together of phase boundaries. For uncorrelated inputs (Sec. pp3-s3), the $\eta$ axis contains a transition point where rotation symmetry is broken (red dot). The perturbation theory for short-range input correlations (Sec. pp3-s3b) shows that this transition point is continued by a flat phase boundary. The asymptotic rank reduction used for the long-range limit (Sec. pp3-s2) reveals an $R\text{−}T$ boundary far away from the origin. We also know from Sec. pp3-s1c that the $N$ phase is forbidden outside a quarter circle containing the red dot on its contour, and the $R$ phase is forbidden inside it. This leads to predicting an $N\text{−}T$ boundary (orange dashed line) contained within the quarter circle, and an $R\text{−}T$ boundary stretching from the red dot into the long-range regime. The red dot is a triple point of the system.