A spiking neural network model of the Superior Colliculus that is robust to changes in the spatial–temporal input

Previous studies have indicated that the location of a large neural population in the Superior Colliculus (SC) motor map specifies the amplitude and direction of the saccadic eye-movement vector, while the saccade trajectory and velocity profile are encoded by the population firing rates. We recently proposed a simple spiking neural network model of the SC motor map, based on linear summation of individual spike effects of each recruited neuron, which accounts for many of the observed properties of SC cells in relation to the ensuing eye movement. However, in the model, the cortical input was kept invariant across different saccades. Electrical microstimulation and reversible lesion studies have demonstrated that the saccade properties are quite robust against large changes in supra-threshold SC activation, but that saccade amplitude and peak eye-velocity systematically decrease at low input strengths. These features were not accounted for by the linear spike-vector summation model. Here we show that the model’s input projection strengths and intra-collicular lateral connections can be tuned to generate saccades and neural spiking patterns that closely follow the experimental results.


The linear dynamic ensemble-coding model
The saccade trajectory is generated by dynamic weighting of each spike of each recruited neuron with its efferent connection strength ('spike vector') to the downstream saccade brainstem circuitry, and linearly adding the effects from the total population: , < =1 =1 (S1) in which ( − , ) is a spike of neuron n at time τn,s, Npop is the total number of active neurons in the population, Nspk,n is the total number of spikes fired by neuron n, and is the cell's sitespecific, fixed 'spike vector'. The latter is determined by each neuron's efferent synaptic connection strength to the horizontal brainstem circuitry [2] , which we here simply specified in one dimension as: with ∈ [0 − 5] mm the cell's anatomical rostral-caudal location in the SC map, encoding the horizontal saccade amplitude, and a fixed scaling factor that only depends on the assumed cell density (in number neurons/mm). This scaling factor was calibrated for a horizontal saccade of 15 deg. The SC afferent mapping parameters (here: A=3.0 deg, and Bu=1.4 mm) had been estimated from [2,3].
We simulated eye movements in response to the presentation of a single input stimulus at a particular location in the input layer. A target point in visual space at T deg from the fovea was thus mapped to its anatomical position, uT, corresponding to the center of a Gaussian-shaped population in the input layer according to the afferent mapping function of Ottes

The AdEx neuron model
The neurons in the network were implemented in the Brian2 spiking neural network simulator [4] with a time resolution of 0.01 ms. We used adaptive exponential integrate-and-fire (AdEx) neurons, which enables realistic bursting dynamics. The AdEx neuron is a simplification of the biophysical Hodgkin-Huxley model, as it contains only two state variables: the membrane potential, V(t), and the adaptation current, q. The dynamics of the AdEx neuron model are determined by two differential equations in which the parameters have a clear physiological interpretation: Here, C is the membrane capacitance, gL is the leak conductance, EL is the resting potential, η is a slope factor, Isyn,n is the total synaptic input current, τq is the adaptation time constant, a is the subthreshold adaptation constant, VT the threshold potential.
In the SC model neurons, two biophysical parameters of the model depended on the location of the neuron (un) in the motor map, and thus determined the bursting properties of these neurons: the adaptation time constant, τq,n , and the synaptic input current, Isyn,n. The other parameters, C, gL, EL, η, VT and a, were taken location-independent and were optimized such that they ensured an appropriate neural bursting behavior (see Table S1, for the values of all parameters). The AdEx neurons in the input layer all had identical biophysical properties, independent of their location in the input map. Therefore, all SC neuron received similar spiking activity from the input layer across different intended saccades.
Optimal parameter values for the AdEx model neurons, and for all network connections, were obtained from a brute-force algorithm, as there exists no analytical solution for the system.
A spike is triggered whenever the membrane potential, V(t), rapidly grows to infinity. In practice, however, we set a spiking threshold, VT. The slope factor determines the sharpness of the threshold and implements a smooth spike initiation zone instead of a strict all-or-nothing spiking threshold. For each spiking event at time, τ, the integration of the equation is reset to its resting potential, Vr, and the adaptation current, q, is increased by b to implement the spike-triggered adaptation:

Synaptic function of the SC neural model
A SC neuron in the motor map receives total input from spiking activity of its surrounding collicular neurons through conductance-based synapses from the excitatory and inhibitory synaptic transmission, and from spiking activity of the input layer neurons, which in turn are stimulated by an external input current. The excitatory-inhibitory synaptic input to an SC neuron is given by: where gexc and ginh are excitatory and inhibitory synaptic conductances of neuron n; Ee and Ei are excitatory and inhibitory reversal potentials, respectively (for their values, see Table S1).
The excitatory conductances respond dynamically to the activity of the collicular neurons and the external cortical input. They increase for each presynaptic spike by the excitatory synaptic connection strength with nearby collicular neurons, and for each spike from the cortical input neuron at the same location, and afterwards they decay exponentially, according to: Similarly, the inhibitory conductance increases with each presynaptic spike from SC neurons with an inhibitory connection strength to neuron n, after which it decays exponentially: In Eqns. S8 and S9, τexc and τinh are the excitatory and inhibitory time constants, respectively; is the synaptic strength between input layer neuron n and SC neuron n; , ℎ and , are the intra-collicular excitatory and inhibitory lateral connection strengths between neurons i and n, while τi,s is the timing of the presynaptic spike in the spike train s, ( − , ), projecting from collicular neuron i to neuron n.

The effect of lateral interactions
To establish the functional role of the intra-collicular lateral connections on network performance in response to large changes in the SC input patterns, we here also illustrate the network's behavior without these lateral interactions. Fig S1a shows how the number of active neurons in the SC layer varies with the input population size, when stimulation is applied at target location T=15 deg at different stimulation strengths, corresponding to different population widths (dark-green curve). The number of activated SC cells increases with the input population size up to a level where all 200 SC cells become recruited. Similarly, the total number of spikes emitted by the SC population appears to increase approximately linearly with the population size (light green). This means that the resulting saccade amplitude (Eq. S1), which was tuned to 15 deg for the default population size of 0.5 mm, increases to 27 deg for the input population size of 1.0 mm. Only the maximum number of spikes emitted by the central cell remained constant, because the input firing rates had been limited in the simulations to 400 spikes/s, and the central neuron receives its input spiking activity only from the neuron in the input layer directly above it, through the one-to-one synaptic connections between the two layers (see also Methods, and Fig 1). Fig S1b shows the results of the network model for the same input stimulation conditions, but now with the lateral interactions as described by Eqns. 2-4. For small input populations (up to 0.3-05 mm), the number of recruited cells, the total number of spikes, as well as the number of spikes of the central neuron, all gradually increase with the input population size. However, the number of activated neurons and the total number of spikes from the motor map increase very little beyond the default stimulation input of 0.5 mm. As a result, the saccade amplitude now follows a saturating relationship with the input stimulation intensity (see also below).

Model output for the default input strength
In Fig S2 we quantified the collicular bursts in response to the external input current with default parameters: σpop = 0.5 mm, I0 = 3.0 pA, and β = 0.03 s -1 , when applied at different sites in the input map. Appropriate tuning of the biophysical parameters of the SC cells, such as τq,n and along the rostral-caudal axis in the SC map, ensured burst profiles that faithfully reflect the experimentally observed spatial variations in the saccade-related SC bursts [5]. Fig S2a shows how the evoked collicular bursts of the central cells in the recruited population systematically reduce their peak firing rates, and increase their duration, with increasing saccade amplitude. Panel S2b quantifies these physiological properties as function of target eccentricity. According to the linear ensemble-coding model (Eqn. S1), these burst properties underlie the nonlinear kinematic mainsequence properties of the associated saccadic eye movements. The resulting saccades and their main-sequence kinematic properties are illustrated in panels S2 c,d for eight stimulation sites in the motor map. Note that the saccade duration increased with the saccade amplitude, and that the peak eye velocity increases in a nonlinear way with saccade size. A further characteristic property of the saccades concerns the skewness of their velocity profiles. Typically, the time-to-peak velocity of saccades changes little across a wide range of saccade amplitudes, while the saccade duration increases systematically with the saccade amplitude.