^{*}

Edited by: Misha Tsodyks, Weizmann Institute of Science, Israel

Reviewed by: Andre Longtin, University of Ottawa, Canada; Paolo Del Giudice, Italian National Institute of Health, Italy

*Correspondence: Zachary P. Kilpatrick, Department of Mathematics, University of Houston, 651 Phillip G Hoffman Hall, Houston, 77204, TX, USA e-mail:

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Competitive neural networks are often used to model the dynamics of perceptual bistability. Switching between percepts can occur through fluctuations and/or a slow adaptive process. Here, we analyze switching statistics in competitive networks with short term synaptic depression and noise. We start by analyzing a ring model that yields spatially structured solutions and complement this with a study of a space-free network whose populations are coupled with mutual inhibition. Dominance times arising from depression driven switching can be approximated using a separation of timescales in the ring and space-free model. For purely noise-driven switching, we derive approximate energy functions to justify how dominance times are exponentially related to input strength. We also show that a combination of depression and noise generates realistic distributions of dominance times. Unimodal functions of dominance times are more easily told apart by sampling, so switches induced by synaptic depression induced provide more information about stimuli than noise-driven switching. Finally, we analyze a competitive network model of perceptual tristability, showing depression generates a history-dependence in dominance switching.

Ambiguous sensory stimuli with two interpretations can produce perceptual rivalry (Blake and Logothetis,

Several principles govern the relationship between the strength of ambiguous stimuli and the mean switching statistics in perceptual rivalry (Levelt,

Most theoretical models of perceptual rivalry employ two pools of neurons, each selective to one percept, coupled to one another by mutual inhibition (Matsuoka,

In light of these observations, we wish to consider the role adaptive mechanisms play in properly sampling ambiguous stimuli in a mutual inhibitory network. Two stimuli of different orientations are presented to the network (Levelt,

Using parameterized models, we will explore how synaptic depression improves the ability of a network to extract stimulus contrasts. First, we study how much information can be determined about the contrast of each of the two percepts of an ambiguous stimulus. In the case of a

As a starting point, we consider a model for processing the orientation of visual stimuli (Ben-Yishai et al., _{m}. Synaptic interactions are described by the integral term
_{0} controls the mean of each peak and _{a} controls the level of asymmetry between the peaks. Effects of noise are described by the stochastic process 〈ξ(

We assume units of time _{m} = 1 (10 ms). Experimental observations have shown synaptic resources specified

We also study space-free competitive neural networks with synaptic depression (Shpiro et al., _{j}(_{j}_{j} and the resource recovery timescale is τ. Fluctuations are introduced into population _{j} with 〈_{j} (_{j}(_{j}(_{j}(0) are initialized by randomly drawing from a uniform distribution on [0, 1]; _{j}(0) are initialized by randomly drawing from a uniform distribution on [1/(1 + β), 1].

The spatially extended model (Equation 1) is simulated using an Euler–Maruyama method with a timestep ^{−4}, using Riemann integration on the convolution term with 2000 spatial grid points. A population is considered dominant if the peak of its activity bump is higher than the other; switches occur when the other bump attains a higher peak. The reduced network (Equation 6) was also simulated using Euler–Maruyama with a timestep ^{−6}. Population _{j} > _{k} (

To generate the theoretical curves presented for exponentially distributed dominance times, we simply take the mean of dominance times and use it as the scaling in the exponential (Equation 28). For those densities that we presume are gamma distributed, we solve a linear system to fit the constants _{1}, _{2}, and _{3} of
^{n}(^{n}) along with its associated dominance times: (^{n}_{1}, ^{n}_{1}); (^{n}_{2}, ^{n}_{2}); (^{n}_{3}, ^{n}_{3}) where ^{n}_{j} = ^{n}(^{n}_{j}). We always choose ^{n}_{2} = arg max_{T} ^{n}(^{n}_{1} = ^{n}_{2}/2 and ^{n}_{3} = 3^{n}_{2}/2. It is then straightforward to solve the linear system

We now present results that reveal the importance of synaptic depression in preserving information about bimodal stimuli. No previous work, to our knowledge, has studied how activity in a ring model with depression (Equation 1) can be collapsed to a low dimensional oscillation. The oscillation results from a combination of depression and mutual inhibition, which produces population dominance times and can thus be sampled to give information about the strength of the stimulus that produced them. Once noise is added to these low dimensional oscillations, dominance time distributions still remain relatively tight, which can be sampled to infer relative contrasts of each input. We contrast this with a previous cue orientation selective model which used a heterogeneous population of spiking neurons with lateral inhibition and slow adaptation, so chaos rather than noise produced apparent stochasticity in dominance times (Laing and Chow,

To start we consider the ring model with depression (Equation 1) in the absence of noise, so ξ ≡ 0. In previous work, noise-free versions of Equation (1) have been analyzed to explore how synaptic depression can generate traveling pulses (York and van Rossum,

We now look for winner-take-all solutions, as shown in Figure _{t} = _{t} = 0, implying

_{0} = 0.6) defined by the bump half-width _{0} = 0.84); _{0} = 1), where initial condition is (Equation 16). Other parameters are κ = 0.5, β = 1, and τ = 50.

Therefore, by simplifying the threshold condition,

The implicit Equation (11) can be solved numerically using root finding algorithms. For symmetric inputs (_{a} = 0), we can solve (Equation 11) explicitly

With this solution, we can relate the parameters of the model to the existence of the winner-take-all state. To do so, we need to look at a second condition that must be satisfied,

At the point in parameter space where the Equation (14) is violated, a bifurcation occurs, so the winner-take-all state ceases to exist. This surface in parameter space is given by the equation

Experiments on ambiguous stimuli have shown sufficiently strong contrast rivalrous stimuli can be perceived as a single fused image (Blake,

Computing the integrals, we find

Oscillations can occur, where the two bump locations trade dominance successively (Figure

_{0} increases both peaks; increasing _{a} decreases the left and increases the right peak. _{R} ≈ 0.9 s) is longer than left (blue bar : _{L} ≈ 0.6 s) for asymmetric input in _{a} = 0) bimodal input (Equation 5) decreases the dominance time _{a} ≠ 0), we find that varying _{R} = _{0} + _{a} while keeping _{L} = _{0} − _{a} fixed changes the dominance times of the left percept _{L} (blue) much more than that of the right percept _{R} (red). Other parameters are κ = 0.5, β = 1, and τ = 50.

To study oscillations, we assume that the timescale of synaptic depression τ » τ_{m}, is long enough that we can decompose (Equation 1), with ξ ≡ 0, into a fast and slow system (Laing and Chow, _{a} = 0). This way, we can simply track _{i}(_{0}. We can examine the fast Equation (17), solving for the form of the slowly narrowing right bump during its dominance phase

We solve for the slowly changing width

We can also identify the maximal value of _{i}(_{0} which still leads to the right bump suppressing the left. Once _{i}(_{0}, the other bump escapes suppression, flipping the dominance of the current bump. This is the point at which the other bump of Equation (20) rises above threshold, as defined by the equation _{0} − _{0} sin(2_{0}) = κ. Combining this with Equation (21) and solving the resulting algebraic equation, we find

The amplitude of synaptic depression is excluded from Equation (22), but we know _{0} ∈ ([1 + β]^{−1}, 1). This establishes a bounded region of parameter space in which we can expect to find rivalrous oscillations, which we use to construct a partitioning of parameter space in Figure

_{0}) into various stimulus-induced states of (Equation 1) when ξ ≡ 0, κ = 0.5, and τ = 50

In the case of an asymmetric bimodal input (_{a} > 0), we can also solve for explicit approximations to the dominance times of the right _{R} and left _{L} populations. Following the same formalism as for the symmetric input case
_{±} = β ± (1 + β) (_{R} − _{L}) and _{R, L} = 4(1 + β) (1 − _{L, R})[(1 + β)_{R, L} − 1], in terms of the local values _{L} and _{R} of the synaptic scaling in the right and left bump immediately prior to their suppression. Notice when _{L} = _{R}, then _{d} = 0 and Equations (23) and (24) reduce to Equation (19). We now need to examine the fast Equation (17) to identify these two values. This is done by generating two implicit equations for the half-width of the right bump _{R} and _{R} at the time of a switch
_{L} = _{0} − _{a} is the strength of input to the left side of the network. Likewise, we can find the value of the synaptic scaling in the left bump immediately prior to its suppression
_{R} = _{0} + _{a} is the strength of input to the right side of the network. Using the expressions (25) and (26) we can now compute the dominance time formulae (23) and (24), showing the relationship between inputs and dominance times in Figure _{R} has a very weak effect on the dominance time of the right percept. Thus, dominance times obey the classic description of Levelt's second proposition (Levelt,

Finally, we demonstrate how the strength of a symmetric input _{0} and strength of depression β lead to different behaviors of the network (Equation 1) in Figure _{0} for which rivalrous oscillations exist. When synaptic depression is sufficiently strong, the range of _{0} that leads to a winner-take-all state narrows. For sufficiently strong _{0}, increasing β leads to a network that reveals a piece of the stimulus that would otherwise be kept hidden. As we will show, synaptic depression helps the network reveal stimulus information in a way that is much more reliable than noise.

We will now study rivalrous switching brought about by fluctuations. In particular, we ignore depression and examine the noisy system

_{0} = 0.9 and _{a} = 0 in bimodal input (Equation 5). _{0} = 0.9. Other parameters are κ = 0.5 and ε = 0.04.

We now explore the task of discerning the relative contrasts of the two stimuli _{R} and _{L} based on samples of the dominance time distributions. Notice in Figure _{R} > _{L} approaches 1/2 as the number of observations _{R} > _{L} | ^{*}(_{R} > _{L} based on sampling dominance time pairs from ^{*}(^{(1)}_{R}, ^{(1)}_{L}; ^{(2)}_{R}, ^{(2)}_{L}; …; ^{(n)}_{R}, ^{(n)}_{L}}. As _{R}(_{R}) = _{L}(_{L}) = _{R} > _{L} | T^{*}(∞)) = 1/2, as in Figure

_{R} is higher than the left input _{L}, based on the sampling _{L} = _{R} = 0.9_{R} > _{L} | ^{*}(

We explore this further in the case of asymmetric inputs, showing dominance times are still specified by exponential distributions as shown in Figure _{R} > _{L}, the exponential distributions _{R}) and _{L}) still have substantial overlap, so sampling from these distributions can yield _{R} < _{L}. Using such a sample to guess the ordering of amplitudes _{R} and _{L} would yield _{R} < _{L}, rather than the correct _{R} > _{L}. In terms of conditional probabilities, we expect situations where _{R} > _{L} | ^{*}(_{R} > _{L}. We can quantify this effect numerically, as shown in Figure

_{R} = 0.92 and _{L} = 0.88, leads to longer dominance times for right percept _{R}. _{R} > _{R}|^{*}(_{R} is stronger than left _{L} based on _{R} and _{L} sampled. Upper gray line is theoretical prediction (Equation 29) of the limit _{L}〉 ≈ 0.5 s) and right (〈_{R}〉 ≈ 1 s) percepts. _{R}〉 and 〈_{L}〉 on the strength of the right input _{R} when _{L} = 0.9. Black curves are best fits to exponential functions of _{R}. _{R} > _{L} | ^{*}(∞)] right input _{R} is stronger than left _{L} in the limit of high sample number

Using Equation (29), we can estimate the limit _{R} > _{L}| ^{*}(∞)) (Figure

We also see the mean dominance times still obey Levelt's propositions (Figure _{R}〉 and 〈_{L}〉 provides very precise information about the ordering of contrasts _{R} and _{L}. However, when comparing successive dominance times, accurately discerning the relative input contrasts is more difficult. This becomes more noticeable when the input contrasts are quite close to one another, as in Figure

We now study the effects of combining noise and depression in the full ring model of perceptual rivalry (Equation 1). Numerical simulations of Equation (1) reveal that noise-induced switches occur robustly, even in parameter regimes where the noise-free system supports no rivalrous oscillations, as shown in Figure _{j} = _{R} being greater than _{L} is a sigmoidal function of _{R} whose steepness increases with β. For no noise, the likelihood function is simply a step function _{R} > _{L}), implying perfect discernment.

_{a} > 0). (A)_{R} = 0.92 and _{L} = 0.88, which leads to right percept dominating longer. _{L}(_{L}) over 1000 s is well fit by a gamma distribution (Equation 30). _{R}(_{R}) across 1000 s is well fit by a gamma distribution (Equation 30). Other parameters are κ = 0.5, β = 0.2, τ = 50, and ε = 0.01.

_{R} > _{R}|^{*}(∞)] the right input _{R} is stronger than the left _{L} based in the limit of an infinite number of samples of the dominance times _{R} and _{R} for the parameters: β = 0, ε = 0.04 (pink); β = 0.2, ε = 0.01 (magenta); and β = 0.4 and ε = 0.0025 (red). Other parameters are τ = 50 and κ = 0.5.

We now perform similar analysis on a reduced network model (Equation 6) and extend some of the results for the ring model. We can construct an energy function (Hopfield,

First, we note Equation (31) has a stable winner-take-all solution in the _{j} > 0 and _{k} < 1/(1 + β) (_{L}, _{R} > 1/(1 + β). Coexistent with the fusion state, there may be rivalrous oscillations, as we found in the spatially extended system (Equation 1). To study these, we make a similar fast-slow decomposition of the model (Equation 31), assuming τ » τ_{m} to find _{j}'s possess the quasi-steady state
_{j} = 0 or 1 almost everywhere. Therefore, we can estimate the dominance time of each stimulus using a piecewise equation for the slow subsystem

Combining the slow subsystem (Equation 33) with the quasi-steady state (Equation 32), we can use self-consistency to solve for the dominance times _{R} and _{L} of the right and left populations. We simply note that switches will occur through escape, when cross-inhibition is weakened enough by depression such that the suppressed population's (_{j} = _{k}. Using Equation (33), we find
_{±} = β ± (1+β) [_{R} − _{L}] and _{R, L} = (1 − _{R, L}) (1 + β) [(1 + β) _{L, R} − 1]. For symmetric stimuli, _{L} = _{R} =

_{L} and _{R} as a function of right input _{R} keeping _{L} = 0.8 fixed as computed by theory (curves) in Equations (34) and (35) fits numerically computed (dots) very well.

Next, we show that the network with depression and noise generates activity oscillations with dominance times that are gamma distributed (Fox and Herrmann, _{m} by assuming we can augment the energy of the depression-free (β = 0) network (Hopfield, _{R} and _{L} (Mejias et al.,

_{R} (black) and _{L} (blue) stay close to attractors at 0 and 1, aside from depression or noise induced switching. Depression variables _{R} (red) and _{L} (green) slowly exponentially change in response to the states of _{R} and _{L}. _{R} = 0.82 and _{L} = 0.78, sampled over 1000 s. The right population has a longer mean dominance time. Other parameters are β = 0.2, τ = 50, and ε = 0.036.

A similar energy function was previously used in a model with spike frequency adaptation (Moreno-Bote et al., _{R} and _{L} change. The energy difference between the right dominant state and fusion is

Notice that dominance times of stochastic switching (Figures

Finally, we will compare the transfer of information in competitive networks that process more than two inputs. Recently, experiments have revealed that perceptual multistability can switch between three or four different percepts (Fisher,

_{1}, _{2}, _{3} and the second synaptic scaling variable _{2} (cyan) of the three population network (Equation 37) driven by symmetric stimulus

We study perceptual tristability in a competitive neural network model with only depression, to start, with a Heaviside firing rate (Equation 4), and symmetric inputs _{1} = _{2} = _{3} =

We are interested in rivalrous oscillations, which do arise in this network (Figure _{m} to compute the dominance time

Now, we study how noise alters the switching behavior when added to the deterministic network (Equation 37). Thus, we discuss the three population competitive network with noisy in activity
_{j} are identical independent white noise processes with variance ε. In Figure

_{f} of a switch being in the forward direction in simulations of (Equation 38) as a function of the amplitude

Mechanisms underlying stochastic switching in perceptual rivalry have been explored in a variety of psychophysical (Fox and Herrmann,

We have studied various aspects of competitive neuronal network models of perceptual multistability that include short term synaptic depression. First, we were able to analyze the onset of rivalrous oscillations in a ring model with synaptic depression (York and van Rossum,

We also used energy methods in reduced models to understand how a combination of noise and depression interact to produce switching. Using the energy function derived by Hopfield (

Mutual inhibitory rate models with terms representing only spike frequency adaptation (Wilson,

Spatially extended neural field models are a useful tool for understanding complex dynamics that emerge in networks connected by synapses that are stimulus preference dependent (Wilson and Cowan,

Note to analytically study the relationship between dominance times and input contrast in the noisy system, we resorted to a simple space-clamped neural network. In future work, we plan to develop energy methods for spatially extended systems like Equation (27). Such methods have seen success in analyzing stochastic partial differential equation models such as Ginzburg-Landau models (E et al.,

The author declares that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.