VENTAJAS COMPETITIVAS DE COLOMBIA Y CENTROAMÉRICA

We use a two-population neural field model extended in space with periodic boundary con- ditions, where the spatial variable x can be in R1 for the network on the ring or in R2 for the network on the torus. A schematic showing the connectivity is shown in Fig 14. The equations are:

τe u(x, t)t = −u + fe(Jee∗ u − Jie∗ v + q S(x; k))

τi v(x, t)t= −v + fi(Jei∗ u − Jii∗ v + q r S(x; k))

(3.1)

where u and v are excitatory and inhibitory synaptic activities, respectively, and provide a low-pass filter on the firing rate, τe and τi are the timescales associated with the excitatory

and inhibitory synapses, respectively, “∗” denotes spatial convolution, and Jβα = aβαKα for

Ke,i= 1 2σe,i √ π exp  −|x|2 2σ2 e,i 

is a gaussian kernel, S(x; k) is our stimulus, parameterized by the wavenumber k and generally taking the form cos(2π k x_N ), where N is the number of excitatory or inhbitory neuronal populations being modeled for the network on the ring, or the square root of the number of excitatory or inhibitory populations in for the network on the torus, q modulates the stimulus strength to both u and v, while r is fixed in our study at 0.8 and represents the inhibitory-to-excitatory stimulus strength ratio. Letting

f (w) = 1

1 + exp(−4w), the firing rate functions, fe,i can be either

fe,i(w) = f (w − θe,i) or (3.2)

fe,i(w) = f (w − θe,i) − f (−θe,i), (3.3)

where θe,i are the excitatory and inhibitory thresholds, respectively. The advantage of the

first formulation is that the activity variables u, v are always positive, resulting in values that can be more directly interpreted physically. The advantage of the second formulation is that it sets the steady state of our network, Eq (3.1), to (u, v) = (0, 0), facilitating further calcualtions. In this formulation, we interpret (0,0) as the baseline firing rate, and a positive or negative values of u or v as an increase or decrease from this baseline, respectively. Importantly, we can easily go back and forth between these two systems using the a simple change of variables. For this change of variables, how we choose the parameters, and the parameter values used, see Sec 3.4.1. While most of the parameters will remain fixed, we vary the values of aee, k, and q to see the effects on the network’s dynamics.

3.1.2 Outline

The organization of the chapter is as follows. First, we look at our model in two spatial dimensions (i.e., the network is on the torus), approximating a sheet of cortex receiving full-field visual stimuli. We examine both simple cosine stripes and more complex noisy

images as stimuli. With both simple and complex stimuli, we find that our model demon- strates spatiotemporal resonance, showing spatially heterogeneous oscillatory activity with weak stimuli for some wave numbers, requiring stronger stimuli for others, and showing no sustained oscillatory activity outside of a narrow band of wave numbers.

We then look at simpler modeling assumptions to gain insight into the network’s resonant responses. First, we restrict our model to be extended in only one spatial dimension (i.e., the network is on the ring), providing a more amenable formulation for further analysis. Using the numerical tool XPP-AUTO [75], we can analyze the network’s dependence on various parameters, defining the regions of instability as functions of, e.g., the recurrent excitation and spatial frequency of the stimulus. In particular, using similar parameters as for the network on the torus, we find the same qualitative responses as on the torus across different stimulus spatial frequencies, demonstrating the utility of reducing our spatial dimensions.

We also study the behavior of the network in response to partial-network stimuli; that is, to stimuli that only project to part of the network. We find parameter regimes that show the spread of large-amplitude oscillatory activity, dynamics that may be highly relevant to PnSE. We then study the network on the ring with stimuli involving the superposition of two spatial modes, allowing for both a more realistic scenario and a still-tractable system to analyze numerically.

Using a perturbation calculation, we can then compute the stability boundaries found above (as a function of the network’s recurrent excitation, aee, and the stimulus amplitude, q,

for different stimulus wavenumbers k) near the onset of instability for small-amplitude stimuli of different wave numbers, finding very good agreement with the numerically-computed regions. These calculations lend further insight into the network’s responses to different stimuli. We review the results and relate them to the relevant biological and psychophysical findings in Sec 3.3, before finally detailing the protocols used to produce the results in Sec

3.2 RESULTS

The network extended in both two and one spatial dimensions with periodic boundary condi- tions imposed (i.e., on a torus and the ring, respectively) displays a strong spatial resonance, showing high sensitivity to stimuli with modes in a narrow band and low or no sensitivity to modes outside of this band, consistent with experimental findings in contrast sensitivity, PnSE, and visual discomfort.

The network on the torus and the ring both give rise to rich and interesting spatial and spatiotemporal patterns that may be relevant to experimental setups that are able to attain sufficient spatial resolution, as well as of mathematical interest. Complex patterns can form and evolve as the result of the interplay of wavelike and more spatially-isolated oscillatory activities. We thus describe the spatial and spatiotemporal patterns that we observe, but we leave a more thorough analysis of such behaviors as future work.