Mean-field analysis of orientation selectivity in inhibition-dominated networks of spiking neurons

Abstract

Mechanisms underlying the emergence of orientation selectivity in the primary visual cortex are highly debated. Here we study the contribution of inhibition-dominated random recurrent networks to orientation selectivity, and more generally to sensory processing. By simulating and analyzing large-scale networks of spiking neurons, we investigate tuning amplification and contrast invariance of orientation selectivity in these networks. In particular, we show how selective attenuation of the common mode and amplification of the modulation component take place in these networks. Selective attenuation of the baseline, which is governed by the exceptional eigenvalue of the connectivity matrix, removes the unspecific, redundant signal component and ensures the invariance of selectivity across different contrasts. Selective amplification of modulation, which is governed by the operating regime of the network and depends on the strength of coupling, amplifies the informative signal component and thus increases the signal-to-noise ratio. Here, we perform a mean-field analysis which accounts for this process.

Keywords: Common-mode attenuation; Contrast invariance; Inhibition-dominated; Mean-field analysis; Operating regime; Orientation selectivity; Tuning amplification.

Figure 1

Raster plot of network activity. (A) Typical raster plot, shown is 1 second of the activity of a balanced network with recurrent synaptic couplings of amplitude EPSP = 0.1 mV, in response to a stimulus of orientation θ = 90° at a medium contrast (s _B= 16 000 spikes/s). Spikes of excitatory and inhibitory neurons are plotted in red and blue, respectively. The plot on the right shows the average firing rate of neurons, computed from the spike count during 6 s of simulation. The histogram on the bottom depticts the time resolved firing rates of the excitatory (red) and the inhibitory (blue) populations, respectively, using a time window of 10 ms. The histogram on the bottom right shows the probability density function of time averaged firing rates of individual excitatory (red) and inhibitory (blue) neurons in the network, respectively. (B) Same spike trains as above, with neurons sorted according to their input preferred orientations (indicated on the y-axis). As above, the plot on the right indicates the firing rate of each neuron computed from the spikes emitted during the simulation.

Figure 2

Tuning amplification and contrast invariance of neuronal responses in the network. (A) The random network (center), composed of excitatory (red, 80%) and inhibitory (blue, 20%) neurons, transforms weakly tuned input (green) to highly selective responses (red). The sample tuning curves are shown for unit # 4715, which is an excitatory neuron. Neurons have very similar tuning curves for different contrasts, and they are indistinguishable after normalization by their respective peak value (see inset). EPSP = 0.1 mV. (B) Spikes elicited by the same excitatory neuron as in (A) in response to different orientations of a stimulus at a medium contrast. (C) More samples of output tuning curves from the network, in polar representation. Radial axis encodes the firing rate of the respective neuron at each orientation, indicated by the angle, with the maximum firing rate indicated as a number next to each plot. Both excitatory (red) and inhibitory (blue) neurons are highly selective in their responses, despite their weakly selective inputs (green) and recurrent inputs of mixed selectivities. Lighter colors correspond to tuning curves for lower contrasts, respectively.

Figure 3

Orientation selectivity across the population. (A) Distribution of a global measure of orientation selectivity, OSI=1−Circular Variance (Ringach et al. 2002), in the network. Lighter colors show the distributions for lower contrasts, respectively. All inputs have an OSI of 0.05. (B) Distribution of an alternative measure of orientation selectivity often used by experimentalists (Niell and Stryker 2008). OSI* is the difference of activity at preferred and orthogonal orientations, normalized by their sum, (r _pref − r _orth)/(r _pref + r _orth). (r _pref and r _orth) are obtained from the best fit of a cosine function to output tuning curves, evaluated at Output PO and Output PO+90°, respectively. Alternatively, OSI* can be computed from a linear interpolation of data points (inset). Lighter colors show the distributions for lower contrasts, respectively. All inputs have an OSI* of 0.1. (C) The OSI of all neurons for medium contrast (MC) vs. low contrast (LC), and for high contrast (HC) vs. medium contrast are plotted in the left and right panels, respectively. The diagonal line indicates a perfect contrast invariance of OSI. (D) Output PO vs. Output OSI for all neurons in the presynaptic pool of the neuron shown in Figure 2A. A stimulus of medium contrast has been applied. The neuron receives input from presynaptic neurons that are themselves highly selective on average (OSI distribution on the right), and which uniformly cover the whole range of possible Output POs (distribution on top). The Output OSI and Output PO of the target neuron are 0.65 and 105°, respectively. Other neurons receive similarly heterogeneous inputs (not shown).

Figure 4

Selective processing of baseline and modulation. (A) General reduced circuit model for the operation of a network on its inputs. (B) Reduced circuit model for selective operation of the network on baseline and modulation components of an input vector. (C) Top: Eigenvalue distribution of the weight matrix, , shown for EPSP = 0.1 mV (J = τ _syn e EPSP = 0.136 mV). For normalization, each entry is divided by the reset potential, V _reset = 20 mV. The ‘exceptional eigenvalue’ (green) corresponds to the uniform eigenmode, i.e. the baseline, and the bulk of the spectrum (orange) determines the response of the network to perturbations of a uniform input. Bottom, left: Eigenvalue distribution for the matrix , which gives the stationary firing rates. Bottom, right: Sorted magnitudes of eigenvalues of and . (D) Baseline and modulation components for individual neurons in the network with EPSP=0.1 mV. Scatter plot (center) shows the modulation vs. baseline component of output tuning curves for all neurons of the network. Baseline and modulation are taken as the mean (F0) and the second Fourier component (F2) of individual tuning curves, respectively. The markers (cyan crosses) show the center of mass of each cloud. The histograms indicate the marginal distributions of baseline (green, top) and modulation (orange, right) components, respectively.

Figure 5

Network gains for the selective processing of baseline and modulation. (A) Normalized baseline gain, (green), and modulation gain, (orange), for a network with a fixed connectivity, but different strengths of recurrent synaptic couplings. Shaded regions represent mean±std. Lighter colors correspond to lower contrasts. (B) As a result of selective attenuation of the baseline, the normalized modulation-to-baseline ratio () is generally much larger than 1. The degree of recurrence of the network presented in Figures 1, 2, 3 and 4 is marked by dashed lines.

Figure 6

Stable dynamics of a network with an unstable eingenvalue spectrum. (A) Eigenvalue distribution of the weight matrix, , shown for EPSP = 0.2 mV (J = 0.27 mV). The same normalization as in Figure 4C is employed, i.e. each entry is divided by the reset potential, V _reset = 20 mV. (B-C) Same plots as in Figure 1A, B, for EPSP=0.2 mV.

Figure 7

Qualitativly similiar functional properties of a network with an unstable spectrum. (A-B) Same distributions of OSI and OSI* as in Figure 3A, B, for a network with EPSP = 0.2 mV. (C) Contrast invariance of the OSI measure, as in Figure 3C, for EPSP = 0.2 mV. (D) Distribution of baseline (F0) and modulation (F2) component in a network with EPSP = 0.2 mV. Labeling is the same as in Figure 4D.

Figure 8

Theoretical prediction of baseline and modulation gains. (A, B) The mean value of baseline (A) and modulation (B) components for each network at each contrast obtained by numerical simulation (circles), along with the values predicted by the theory decribed in the text (solid lines).

Figure 9

Non-interference of baseline and modulation. (A) Mean and standard deviation of baseline (green) and modulation (orange) components for the medium contrast are plotted in logarithmic scales for comparison. As in Figure 8, dots and solid lines indicate simulated and predicted values, respectively. Shadings indicate mean±std. As in Figure 5, mean and standard deviation are evaluated over all neurons in the network. (B) The product of the weight matrix with a baseline, , and a modulation input vector, . The entries of the baseline vector are all normalized to one, i.e. the input to each neuron is 1. The operation of the weight matrix on the baseline vector, , is plotted in green. The operation of the weight matrix on the modulation vector, , for three different orientations (θ = 30°,90°,120°) are plotted in magenta, black and cyan, respectively. The corresponding distributions of individual responses are plotted in the magnified histograms on the right. Note that the assumption of ‘perfect balance’ implies a very narrow distribution around zero. The average (over 12 orientations) of the responses, , are plotted in orange. For 12 sample neurons from the network the response vs. orientation of the stimulus are plotted in the inset (center).

Figure 10

Tuning of neuronal inputs. (A) Tuning of the mean input from excitatory (red) and inhibitory (blue) neuron populations, and of the external input (green). Shown are four sample neurons, for stimulations at the medium contrast. EPSP = 0.2 mV. The total recurrent input (excitatory + inhibitory), and the net modulation of the input (external + total recurrent) are also plotted (in black and orange, respectively). The input is computed by replacing each presynaptic spike by an alpha kernel and computing the mean amplitude of the shot-noise signal (in mV/ms). 1 mV/ms corresponds to a mean membrane potential at the spike threshold of the neurons in our simulations. (B) Same for twelve sample neurons, along with their mean values (thicker lines).

Figure 11

Linear tuning in recurrent networks. (A) Linearized gains for single neurons embedded in the network. The extra firing rate, δ r, of a neuron produced in response to a small perturbation, J _s δ s, in the input intensity, plotted for different baseline inputs corresponding to different contrasts. The response is computed by numerically perturbing the mean-field equations (see Methods). The linear gain, ζ=δ r/(J _s δ s), is then computed by linear regression of data points. For this example with EPSP = 0.2 mV, the value ζ = 0.026 is obtained, which is also used for the next panels. (B) For the sample neuron in Figure 2A, all presynaptic Tuning Vectors are extracted (, weighted by the linear gain (ζ), and vectorially added together, reflecting linear integration in neurons. Although each presynaptic vector makes only a small contribution, the resulting random sum can lead to a large resultant Tuning Vector. These are generally larger for presynaptic inhibition (Presyn. Inh.) compared to presynaptic excitation (Presyn. Exc.). Note different scales of axes. (C) Left panel: The resultant vectors for recurrent excitation (Rec. Exc.), recurrent inhibition (Rec. Inh.), total recurrent (Rec. Tot. = Rec. Exc. + Rec. Inh.), feedforward input (Input), and the total input (Tot. = Input + Rec. Tot.) are plotted. All normalized input Tuning Vectors have the same length of one, denoted by the green circle. Right panel: Total recurrent Tuning Vectors (Rec. Tot.) for all neurons in the network are compared with the normalized length of their input Tuning Vectors (green circle). D) Distribution of the length of all Tuning Vectors for all the neurons in the network. Dashed lines show the predicted distributions of the linear analysis in each case (see text for details).

Figure 12

Input vs. output preferred orientation of neurons in the network. (A-C) Output PO vs. Input PO for three values of recurrence, at the medium contrast. Increasing the recurrence leads to more scatter about the diagonal. For illustration purposes, the excitatory (red) and inhibitory (blue) neurons have been plotted only above or below the diagonal, respectively. (D) To quantify the amount of PO change when going from input to output, the Scatter Degree Index (SDI) is plotted as the angular deviation of Δ PO = OutputPO−InputPO (see Methods). The maximum value of this index is ≈40.5°, which corresponds to a uniform distribution of Δ PO. Darker colors show higher contrasts, respectively.

Figure 13

Neuronal tuning curves in weakly (left) and strongly (right) recurrent networks. (A) Tuning curves of a sample neuron (same as in Figure 2A), for different degrees of recurrence in the network, as indicated by different EPSP amplitudes. (B) More (125) sample tuning curves from the network, aligned to their Input PO. Red and blue curves show excitatory and inhibitory output tuning curves for the medium contrast, respectively. Shown in green is the input tuning curve, normalized to the average (over the population) of the mean (over all orientations) of all tuning curves in the network. (C) Mean and standard deviation (across neurons) of all aligned tuning curves, for networks with different degrees of recurrence. Lighter shadings denote lower contrasts, respectively.

Figure 14

Distribution and contrast invariance of selectivity. (A-C) Distribution of orientation selectivity in three different regimes of recurrence. Lighter colors code for lower contrasts. The mean OSI for each distribution is indicated in the corresponding brightness, respectively. OSI* is computed from the cosine fit, as in the main panel of Figure 3B. (D) The mean OSI of all neurons in the network is shown for different levels of recurrence, at three different contrasts. Lower contrasts are plotted in lighter colors. Increasing the recurrence makes the OSI less susceptible to changes in contrast. For high recurrences, this invariance comes at the expense of a decreased selectivity, as the mean OSI in the network decreases. This trade-off between selectivity and invariance is quantified (in brown) by the average (over contrasts) of mean OSI (across neurons) divided by the standard deviation (over contrasts) of the average OSI (across neurons).

Figure 15

Tuning of the membrane potential. (A) Membrane potential of a sample excitatory neuron from the same network with EPSP = 0.2 mV, at the medium contrast. Traces of the membrane potential are plotted for 1 s of response to the preferred (red) and its orthogonal (cyan) stimulus orientation. The histograms of the membrane potential for 6 s of stimulation are shown on the right. (B) Traces of the membrane potential along with the elicited spikes for 12 orientations, for 1 s of recording. The tuning curves of the mean membrane potential (red) and the corresponding firing rate (black) is computed from 6 s of stimulation in the right panel.

Figure 16

Membrane potential at different contrasts. (A) Left: Tuning of the membrane potential of the same neuron as in Figure 15 for different contrasts. Center: Average tuning curve of 24 (12 excitatory, 12 inhibitory) randomly sampled neurons, ranging over all POs. The tuning curves are aligned at a PO of 90°. Error bars indicate mean±std over sampled neurons. To improve the display, they are plotted only for every third data point. The mean membrane potential (over all neurons and all orientations) is indicated by solid, horizontal lines in each case. The shading represents the standard deviation (over neurons) of the mean (over orientations) of the average membrane potential (v _B). Right: Same as the panel in the center, for the free membrane potential, V _fm =V + τ rV _reset. (B) The same mean±std (over sampled neurons) of tuning curves for the distance to threshold of free membrane potentials (V _fm − V _th), for different regimes of recurrence. For the lowest contrast, most of the error bars are smaller than the size of markers and hence not visible. The mean membrane potential and its standard deviation over the (sampled) population is shown, as before, by horizontal lines and shadings, respectively.

Figure 17

Spiking activity in the inhibition dominated regime. (A) Statistical balance of overall excitatory (red) and inhibitory (blue) input from the recurrent network. The dominant inhibition keeps the net recurrent input (gray) negative on average, and only occasional transients lead for a moment to a net excitatory drive from the network. EPSP = 0.2 mV. (B) Spike triggered averages (STA) of excitatory (Exc.) and inhibitory (Inh.) recurrent inputs are plotted from spikes of 12 randomly sampled neurons in response to one stimulation (6 s). The total recurrent input (Tot.) is plotted in black. The membrane potential is normalized by V _th (Norm. Vm.).

Figure 18

Mean distance to threshold sets the operating point of the network. (A) The mean membrane potential for networks with different EPSP sizes (circles), at different contrasts, along with the predicted values (solid lines). Mean membrane potential is computed as the average (over orientations) of mean tuning curves of 12 sample neurons (the same as in Figure 16A). Lighter colors belong to lower contrasts, respectively. (B) Input modulation normalized by the distance to threshold, Vth−Vm, (solid lines) compared to the output modulation (orange circles; same as Figure 8B). Input modulation is given as the input modulation rate times its efficacy (J _s ms _B).

Figure 19

Linear gains determine the gain and stability of networks in response to modulation. or the medium contrast, linear gain (ζ, black line) is computed at each EPSP and is compared with the corresponding modulation gain, γ _M= Output modulation/Input modulation (orange). ζ is computed as ζ = δ r/(J _s δ s), for a small perturbation of the input, δ s = 100 spikes/s. Inset: The radius, ρ, of bulk spectrum of W, normalized by the linear gain (ζ) at each EPSP. Instead of dividing J by V _th (as in Figure 4C), J is now multiplied by ζ. As a result, the normalized radius is now obtained as .