A Computational Model of Innate Directional Selectivity Refined by Visual Experience

The mammalian visual system has been extensively studied since Hubel and Wiesel’s work on cortical feature maps in the 1960s. Feature maps representing the cortical neurons’ ocular dominance, orientation and direction preferences have been well explored experimentally and computationally. The predominant view has been that direction selectivity (DS) in particular, is a feature entirely dependent upon visual experience and as such does not exist prior to eye opening (EO). However, recent experimental work has shown that there is in fact a DS bias already present at EO. In the current work we use a computational model to reproduce the main results of this experimental work and show that the DS bias present at EO could arise purely from the cortical architecture without any explicit coding for DS and prior to any self-organising process facilitated by spontaneous activity or training. We explore how this latent DS (and its corresponding cortical map) is refined by training and that the time-course of development exhibits similar features to those seen in the experimental study. In particular we show that the specific cortical connectivity or ‘proto-architecture’ is required for DS to mature rapidly and correctly with visual experience.

2 To achieve a spatio-temporal down-sampling of spike input from the Input layer resolution (128x128) to the LGN layer resolution (32x32) the membrane time constant of LGN neurons ( lm  ) is set at 10ms and the refractory period is also 10ms. This ensures that the first firing of any Input neuron in the 4x4 group connected to the LGN neuron will cause the LGN neuron to fire but immediate firing of other Input neurons in the group within the refractory period will not cause additional spikes in the LGN neuron.
Cortical layer neurons were modelled as more complex LIF neurons, based upon the Vogels and Abbott CUBA (CUrrent BAsed) model [2]: where V is the membrane voltage, e g is the contribution from excitatory synapses, methyl-4-isoxazolepropionic acid) receptor model. It is assumed that the action potential generated by the presynaptic neuron is instantaneous and decays exponentially over time [3]. A positive noise term N(t) was added to simulate background noise given by the 1 st order system [4]: where N(t) is the noise at time t, n  is the mean of the noise, where g is the effective conductance for an excitatory or inhibitory synapse, and s  is the synaptic time constant. When a presynaptic neuron fires the effective conductance for excitatory and inhibitory synapses was updated by: Where old g is the original effective synaptic conductance, new g is the updated effective synaptic conductance, and w is the synaptic weight. Parameter values are given in Table S1.

Network Structure Initialization
The network architecture is shown in Figure 1 of the main article. The input layer is connected to the LGN layer with excitatory connections with fixed weights of value 1.0.
These connections were set up such that a 4x4 connection field (CF) from the input layer is connected topologically to 1 neuron in the LGN layer. CFs were not overlapping, thus each neuron in the LGN layer averages the activity from 16 pixels in the Input layer (the box marked 1 in Figure 1 is illustrative). The neuron time constant and refractory period for the LGN layer neurons were set to ensure that any activity in the 4x4 group of input neurons resulted in one spike in the LGN Layer neuron (i.e. no multiple firing).
Similarly, the LGN layer is not fully connected to the Cortical layer, but, in keeping with the approach of previous works modelling the visual system each cortical neuron only 'sees' neurons from the LGN layer within its connection field. The CFs from each cortical neuron overlap: see the box marked 2 in Figure 1 Table S2 for a summary of all the network parameters and their initial values.
The network was implemented using the Brian spiking neural simulator [5].

Calculation of Neuron Preference and Selectivity Index (SI)
The average of the firing rate (number of spikes generated during presentation of a pattern) in response to each directional pattern was calculated by presenting all 10 instances of each direction to the networks and averaging the responses. This data was collected for ten untrained networks, at a point midway in training and after training. Neuron orientation and direction preference was calculated using the vector average method described in [6].
For orientation, firing rates were averaged over the two opposite directions of motion as was done in [7] and the vector sum V(x,y) for each neuron was calculated using equations (S9) and (S10).

 
cos(2* ) where   is the firing rate for orientation  and x V and y V are the x and y component sums.
The preferred orientation  can then be found using equation (S11): Note that equation (S11) produces orientations in the range 0 to +/-180 degrees. To convert to 0-180 range, 180 degrees is added to negative angles.
For direction preference the same method is used except that as direction is 2 -periodic,  is not multiplied by 2 in equations (S9) and (S10) and there is no division by 2 in equation (S11). Negative angles are converted to 0-360 range by adding 360 degrees.
The Selectivity Index (SI) is the magnitude of vector V. Normalised selectivity is calculated using equation (S12).  Tables   Table S1 -  Randomly initialised between 0.4 and 0.5 W lat , lateral synaptic weights Randomly initialised between 0.0 and 0.1 (exc) and -0.1 and 0.0 (inh) Exc_p conn , connection probability for lateral excitatory connections Calculated as exp(-dist/sigma) where dist is the Euclidean distance between the neurons and sigma is 3.5 Inh_p conn , connection probability for lateral inhibitory connections Calculated as exp(-sigma/dist) where dist is the Euclidean distance between the neurons and sigma is 8.0