The authors have declared that no competing interests exist.

Orientation selectivity is a key property of primary visual cortex that contributes, downstream, to object recognition. The origin of orientation selectivity, however, has been debated for decades. It is known that on- and off-centre subcortical pathways converge onto single neurons in primary visual cortex, and that the spatial offset between these pathways gives rise to orientation selectivity. On- and off-centre pathways are intermingled, however, so it is unclear how their inputs to cortex come to be spatially segregated. We here describe a model in which the segregation occurs through Hebbian strengthening and weakening of geniculocortical synapses during the development of the visual system. Our findings include the following. 1. Neighbouring on- and off-inputs to cortex largely cancelled each other at the start of development. At each receptive field location, the Hebbian process increased the strength of one input sign at the expense of the other sign, producing a spatial segregation of on- and off-inputs. 2. The resulting orientation selectivity was precise in that the bandwidths of the orientation tuning functions fell within empirical estimates. 3. The model produced maps of preferred orientation–complete with iso-orientation domains and pinwheels–similar to those found in real cortex. 4. These maps did not originate in cortical processes, but from clustering of off-centre subcortical pathways and the relative location of neighbouring on-centre clusters. We conclude that a model with intermingled on- and off-pathways shaped by Hebbian synaptic plasticity can explain both the origin and development of orientation selectivity.

Many neurons in mammalian primary visual cortex are highly selective for the orientation of visual contours and can therefore contribute to object recognition. Orientation selectivity depends on on- and off-centre retinal neurons that respond, respectively, to light and dark. We describe a signal-processing model that includes both subcortical pathways and cortical neurons. The model predicts the preferred orientation of a cortical neuron from the empirically determined spatial layout of retinal cells. Further, the subcortical-to-cortical connections change in strength during visual development, meaning that cortical neurons in the model have orientation selectivity just as precise as real neurons. Our model can therefore explain the origin of orientation selectivity and the way it develops during visual system maturation.

Response properties in the visual system undergo a remarkable change in the transition from subcortical pathways to cortex. Cortical neurons are selective for stimulus characteristics such as contour orientation, motion direction and depth. In primate and carnivore subcortical neurons, by contrast, these selectivities are weak or absent [

Orientation selectivity is a clear example of the subcortical-to-cortical transformation. Many cortical neurons respond best to a contour with specific orientation (for example, vertical) and less well to other orientations. Orientation selectivity was first described by Hubel and Wiesel [

Parts of this model have survived the test of time. Simultaneous recording of cortical neurons and their subcortical inputs have demonstrated the convergence of on- and off-pathways [

The Hubel and Wiesel model has also, however, encountered significant challenges. A recent study [

A third challenge to the convergence model for orientation selectivity comes from response amplitude. In the Soodak [

In this paper, we describe a signal-processing model that complies with known anatomy and physiology of the early visual pathways. On-centre and off-centre inputs to a cortical neuron are co-extensive; a Hebbian development process functionally segregates the two signs of input. Intracortical inhibition, which is assumed to derive from a widespread, slow-acting network that indiscriminately reduces membrane potential, contributes to orientation selectivity through the iceberg effect [

To make the modelling manageable, the scope of the model is limited in several ways. First, the model is designed to describe the cat’s visual pathway because the visual literature for this species is particularly rich (including almost all the animal studies cited above). Second, the subcortical pathway is chosen to pass through the X-type retinal ganglion cell because of its relatively high acuity. Last, the model is restricted to monochromatic, monocular stimuli, and the input layers of primary visual cortex. The model builds on a previous one [

A flow diagram of the model is shown in

Our aim in this paper is to describe a physiologically plausible model that reproduces key aspects of orientation selectivity. It has become increasingly clear over recent years that cortical properties depend heavily on the response characteristics of subcortical channels [

Hebbian changes could potentially increase a synapse’s strength beyond the physiological limit. But the model includes intracortical inhibition which is driven by the same geniculocortical input as are the excitatory neurons. As synapses increase in strength so does inhibition: this limits excitatory responses in the cortex, preventing further synaptic strengthening. We now demonstrate this growth in inhibition during development and its effect on response time courses.

To measure orientation tuning in the model we drifted a sinusoidal grating across the visual field at a variety of orientations.

Previous work, particularly optical imaging, has shown that preferred orientation forms characteristic patterns across the cortical surface [

We compared the statistics of the map with published work by calculating its periodicity. Each orientation in

The geniculocortical weight maps in

In this paper we have described a visual system model supporting the following conclusions.

A Hebbian process is sufficient to functionally segregate on-centre and off-centre inputs to primary visual cortex.

This segregation can produce orientation-selective neurons.

The resulting selectivity has a precision mirroring that of real cortical neurons.

The cortical map of orientation preference arises not from cortical sources but from local clustering of on-centre and off-centre neurons in the retina.

This last conclusion leads to two predictions for future experiments. First, measurement of the locations of on- and off-centre β ganglion cells in the cat retina will allow the calculation of the preferred orientations in the corresponding region of primary visual cortex. An experiment testing this prediction would be difficult to perform because it would require simultaneous measurement of retinal arrays and preferred orientation in the cortex. We therefore offer a second prediction, which should be rather more straightforward to test: that the periodicity of the orientation map depends on the spatial profile of the geniculate relay cell’s centre mechanism. This result follows from the analysis illustrated in

A number of previous models have addressed issues such as orientation selectivity and cortical mapping. How do our results fit in with this previous work? One of the earliest studies was by von der Malsburg [

Somers, Nelson and Sur [

One of the advantages of the Somers et al. [

Previous work has shown that orientation preference maps are firmly established early in visual life, even though orientation tuning is weak. Chapman et al. [

Later work, however, has introduced a new factor into this parsimonious picture. Smith et al. [

Previous models have assumed that orientation selectivity depends on discrete lines of on- and off-centre cortical inputs [^{2} [^{2} [

A recent paper [

There are two sources of randomness in the model, both structural. The first is the random variation of a channel location about its node on a rectangular grid. This variation seeds the orientation map. The second source of random variation is the sequence in which channels are tested with an increase in geniculocortical strength. Changing this sequence has little effect on the results. We have chosen not to add noise to the membrane potential: all of the differential equations defining the model are deterministic. This choice comes with two disadvantages. First, two studies [

We calculated impulse rate by thresholding membrane potential without considering random fluctuations in potential. There is strong empirical support for this approach. Carandini and Ferster [

Resting activity is a key component in our model. In keeping with subcortical measurements [

Here we derive the equations describing the model. Each neuron is represented by a single nonlinear differential equation, and time courses are obtained by simultaneous numerical integration of the equations for all neurons. The difference between membrane potential at the initial segment of a neuron’s axon and action potential threshold determines the action potential rate. This difference is therefore called the generator potential, denoted by _{i}(_{i}. The model assumes that the neuron is a low-pass filter that integrates the difference between the driving potential and generator potential:

To complete _{i}, is proportional to presynaptic impulse rate. The general equation for a model neuron is then:
_{i}.

The general equation requires modification for each stage of the model. The stage numbers are 1 to 7 representing, in order, photoreceptors, bipolar cells, ganglion cells, geniculate neurons, inhibitory neuron somas, inhibitory neuron axons, and excitatory cells. _{s} and _{s} are the contrast sensitivity and radius of the centre mechanism (the model does not include a surround mechanism) and (_{j}, _{j}) is the spatial location of channel

Bipolar cells do not produce action potentials and can be on- or off-centre:
_{j} and _{j} are the time constant and sign for channel _{s}, to the driving potential:

The input to inhibitory cortical neuron _{kc} is obtained by subscript substitution into _{jk} is the strength of the synapse from subcortical input

Finally, excitatory cortical neuron

We transformed the model into the frequency domain for two reasons. First, we reduced the possibility of mathematical and computational errors by ensuring that the solutions in the temporal and frequency domains agreed to within round-off error. Second, we reduced computation time by performing most of the calculations in the frequency domain. The Fourier transform of

Similarly, the transforms for the following subcortical stages are:

Most of the simulations in the paper use a drifting grating as stimulus. We made these simulations faster by using the analytical solution for the dot product, _{stim} is the spatial frequency, _{stim} is the temporal frequency, and _{j} being the location of the

A neuron’s location is defined in the model by the centre of the convergence function that weights its inputs. Off-centre subcortical channels were located at the nodes of a square grid aligned with the visual field patch, with a node at the centre of the patch. Each location was then perturbed with a Gaussian deviate in both the horizontal and vertical directions. Jang and Paik [

The model used a retinal ganglion cell map based on the work of Wässle et al. [

The development process adjusted the strength of the synapse of each geniculate neuron onto each of its cortical targets. At the start of development all of these synapses were assigned a weight of 1. For each development cycle a geniculate neuron was selected, with all neurons equally likely to be chosen. All synaptic weights for this neuron were increased by 0.2 and the model was stimulated with gratings drifting in 16 directions evenly distributed across the whole range. Each excitatory cortical neuron’s impulse rate was calculated, and if the maximum response increased relative to the previous cycle, the weight increase was retained for that neuron. Otherwise, the weight was reduced by 0.2 relative to its value on the previous cycle. Weights were restricted to lie between 0 and 2.

The number of development cycles was determined as follows. Each geniculocortical synapse needed five cycles to change from its starting value to its minimum or maximum. The number of cycles was therefore set at five times the number of geniculate neurons, 5×3281 = 16405, and rounded to the nearest thousand, 16000. Figs _{e}.

Glossary of model parameters and their values to two significant figures.

Symbol | Parameter | Value | Unit |
---|---|---|---|

Contrast | 0.3, unless otherwise stated | None | |

_{s} |
Contrast sensitivity of centre mechanism | 62 | mV / contrast-unit |

_{c} |
Gain of geniculocortical convergence | 3.5 | None |

_{e} |
Gain of inhibitory-excitatory convergence | 2.2 | None |

_{rect} |
Rectifier gain | 7.2 | Hz/mV |

Index of subcortical channel | 1,2,…,3281 | None | |

Index of cortical neuron | 1,2,…,6561 | None | |

_{j} |
Sign of subcortical channel |
None | |

Temporal frequency | Variable | radians/s | |

_{stim} |
Stimulus temporal frequency | 2π × 2 | radians/s |

Generator potential, or difference between membrane and resting potential | Variable | mV | |

_{s} |
Static subcortical depolarisation | 1.9 | mV |

_{s} |
Radius of centre mechanism | 0.4 | deg |

_{c} |
Radius of geniculocortical convergence | 0.95 | deg |

_{e} |
Radius of inhibitory-excitatory convergence | 0.95 | deg |

Spatial frequency | Variable | radians/deg | |

_{stim} |
Stimulus spatial frequency | 2 |
radians/deg |

Time | Variable | s | |

Time constant, stages 1, 5, 7 | 0.01 | s | |

_{inh} |
Time constant, stage 6 | 0.2 | s |

_{j} |
Time constant of channel |
s | |

Motion direction of stimulus | Variable | radians | |

_{jk} |
Weight of synapse from subcortical channel |
0–2 | None |

Horizontal position in visual field | Variable | deg | |

_{j} |
Horizontal position of subcortical channel |
Variable | deg |

_{k} |
Horizontal position of cortical neuron |
Variable | deg |

Vertical position in visual field | Variable | deg | |

_{j} |
Vertical position of subcortical channel |
Variable | deg |

_{k} |
Vertical position of cortical neuron |
Variable | deg |

Index of processing stage | 1, 2, …, 7 | None |

We simulated a visual field patch centred on the horizontal meridian, and 11° from the central area. The size, 8°×8°, is substantially larger than a typical cortical receptive field.

Hughes [

The mean density of β ganglion cells at 11° eccentricity is 1275 cells/mm^{2} [^{2} = 51 cells/deg^{2}. Wässle et al. [^{2} and, similarly, 26.6 for off-centre cells.

Wässle et al. found that the packing of same-sign β cells ranged between square and hexagonal arrays; we used a square grid for simplicity. They measured the distance between same-sign nearest neighbours and found that the standard deviation divided by distance was 0.189. We therefore placed on-channels on a square grid with

Saul and Humphrey [_{s} = 0.40°.

We use the value measured by Tusa et al. [^{2}/deg^{2}.

This radius is derived from the work of Jones and Palmer [_{c} = 0.95°.

The critical issue in choosing cortical cell density is that it be substantially less than the radius of the geniculocortical convergence. We chose to situate both excitatory and inhibitory neurons on a square grid with 0.1° spacing.

We have not found a measurement of inhibitory receptive field radius as rigorous as the Jones and Palmer analysis of simple cells. We have therefore set the radius equal to _{c}.

Each subcortical channel consists of a cascade of four first-order low-pass filters. The impulse response of a series of

It has recently been shown that off-centre X-type geniculate cells lead their on-centre counterparts. The leading edge of the impulse response in off-cells precedes that in on-cells by a mean of 3 ms when measured at 40% of maximum response [_{j} = 11 ms, on-channel, _{j} = 9 ms, off-channel.

Inhibitory strength is an important variable in our model, but setting this strength involves a trade-off between the inhibitory time constant and the inhibitory-to-excitatory gain. We have chosen to set the inhibitory time constant to a relatively large value and use empirical evidence to set inhibitory-excitatory gain as described below. The time constant, 200 ms, is large enough to match the long-lasting inhibitory tail seen in simple cells responding to flashed stimuli [

The form of the generator function and its gradient, _{rect} = 7.2 Hz/mV, are taken directly from the work of Carandini and Ferster [

This parameter is calculated by integrating the centre mechanism’s spatial profile over both dimensions:

We set this equal to the contrast sensitivity of the X-type ganglion cell centre mechanism, 620 Hz/contrast-unit (from the 2 Hz data in Fig 12 of Frishman et al. [_{s} is given by:

The resting impulse rate of geniculate cells averages 14 Hz [

The contrast sensitivity of cortical neurons is best calculated from empirical measurements of cortical membrane potential, which avoid the complications of action potential thresholding. Carandini and Ferster [_{c} was set so that model contrast sensitivity was close to this value.

This gain sets the resting hyperpolarisation in excitatory cells. Anderson et al. [_{e} was set so that the resting hyperpolarisation in excitatory cells approximated this value.

The subcortical map was represented by a grid with elements fine enough (0.005°×0.005°) that on- and off-channels did not coincide.

On- and off-channels locations were assigned the grid values 1 and –1, respectively, and all other locations zero.

Gabor functions were constructed with a standard deviation of 0.7°, the standard spatial frequency (0.5 cycles/deg), 8 orientations uniformly distributed across the range 0 to 180°, and 8 phases uniformly distributed across the range 0 to 360°.

The dot product of the channel grid and each of the (stationary) Gabors was calculated.

For each grid location the maximum value of the product across orientations and phases was determined.

The orientation that yielded the maximum value at each grid point was used as the preferred orientation.

The periodicity of the orientation preference map shown in

We defined a pinwheel as a location in the orientation preference map for which at least three orientations were contiguous. Counting pinwheels in the central 5°×5° of ^{2}. From the preceding paragraph, the area of an orientation hypercolumn is 1.6^{2} deg^{2}. Multiplying these two values gives 2.8 pinwheels/hypercolumn.

Off-centre ganglion cells in the model lie on a square grid perturbed by Gaussian deviates. On-cells lie on a perturbed grid offset diagonally from the off-centre grid. Given that on- and off-cells have differing densities, aliasing can occur along the diagonal axis. The density of off-cells is 26. 6 cells/deg^{2} so that the distance between neighbouring cells on the diagonal axis averages _{on} = (_{off}. Thus the aliasing period is _{on} = _{on}_{off}/(_{on}−_{off}) = 6.5°.

The receptive field profile shown in ^{2}/_{s}^{2}). Given the densities of the retinal ganglion cells, the distance between neighbouring on- and off-cells will typically be less than 0.1°. This is small relative to the radius of the profile, _{s} = 0.4°. The difference between the on- and off-centre receptive field profiles can therefore be well approximated by the derivative of

All simulations were performed in Matlab 2017b (The MathWorks, Inc): the computer code is provided in the Supporting information. Computational errors were reduced by running the model in both the temporal and frequency domains and ensuring that the solutions matched to within round-off error. The model was simulated using an 8°×8° visual field patch but only 6°×6° is displayed, to reduce edge effects. There were 3281 subcortical channels and 6561 excitatory cortical neurons, and therefore 2.2×10^{7} geniculocortical weights. The weights were calculated over 16,000 development cycles. This calculation, which took 176 machine hours, was performed on the University of Sydney’s high-performance computing cluster,

(ZIP)

We thank Jayson Jeganathan, Blake Segula, Ruby Ly and Peter Nguyen who contributed to earlier versions of this model. We also thank Jose-Manuel Alonso for his valuable comments on a draft of the manuscript.