Abstract

Spike synchrony, which occurs in various cortical areas in response to specific perception, action, and memory tasks, has sparked a long-standing debate on the nature of temporal organization in cortex. One prominent view is that this type of synchrony facilitates the binding or grouping of separate stimulus components. We argue instead for a more general function: a measure of the prior probability of incoming stimuli, implemented by long-range, horizontal, intracortical connections. We show that networks of this kind—pulse-coupled excitatory spiking networks in a noisy environment—can provide a sufficient substrate for stimulus-dependent spike synchrony. This allows for a quick (few spikes) estimate of the match between inputs and the input history as encoded in the network structure. Given the ubiquity of small, strongly excitatory subnetworks in cortex, we thus propose that many experimental observations of spike synchrony can be viewed as signs of input patterns that resemble long-term experience—that is, of patterns with high prior probability.

1  Introduction

Depending on the behavioral context, specific cortical neurons fire in synchrony. In the sensory cortex, this depends on qualities of the sensory input: sounds evoke simultaneous activity in auditory cortex cells with matching receptive fields (Brosch, Budinger, & Scheich, 2002; Atencio & Schreiner, 2013), distant cells in somatosensory cortex synchronize when particular skin regions are stimulated (Reed et al., 2008), and synchrony in the primary visual cortex (V1) varies with geometrical stimulus features such as spatial continuity (Gray, König, Engel, & Singer, 1989; Livingstone, 1996) or similarity of orientation (Kohn & Smith, 2005). This can be observed as soon as 30 ms after a stimulus change (Maldonado et al., 2008), that is, within only a few spikes. Beyond sensory cortex, spike synchrony in the primary motor cortex varies with either the performed (Jackson, Gee, Baker, & Lemon, 2003) or the intended action (Riehle, Grün, Diesmann, & Aertsen, 1997; Pipa, Riehle, & Grün, 2007), and it has has been shown that distant prefrontal cortex cells synchronize selectively during memory tasks (Pipa & Munk, 2011).

Interpretations of such results have been conflicting. They have either been seen as an (inconsequential) epiphenomenon of cortical connectivity (Shadlen & Movshon, 1999) or as evidence for a synchrony-based mechanism to quickly assemble and group different sources of currently relevant information (Roelfsema, Engel, König, & Singer, 1996; Gray, 1999). Regardless of their function, the neural basis of these effects is mostly sought in long-range, horizontal, intracortical connections (Stettler, Das, Bennett, & Gilbert, 2002; Li & Gilbert, 2002). Horizontal connections are known to adapt to experience during development (Galuske & Singer, 1996; Schmidt, Galuske, & Singer, 1999) and in adult learning (Rioult-Pedotti, Friedman, Hess, & Donoghue, 1998). In the case of V1, their structure has been found to reflect the aggregate statistics of natural visual scenes (Onat, Jancke, & König, 2013). Perhaps as a result of such adaptations, the fine-scale topology of these networks is complex and functionally heterogenous (Martin, Roth, & Rusch, 2014; Rothschild, Nelken, & Mizrahi, 2010); connections between cells with differing response properties are common. However, one of the few broad regularities in these networks appears to be that cells that respond in similar contexts tend to be well connected. This holds across many cortical areas. For example, direct horizontal connections are found between distant auditory cortex cells with similar response selectivity (Read, Winer, & Schreiner, 2001) and between primary motor cortex cells representing related muscle groups (Weiss & Keller, 1994). V1 cells with nearby receptive fields are preferentially connected, and even more so where they select for similar visual orientations (Ts'o, Gilbert, & Wiesel, 1986; Stettler et al., 2002). V1 connectivity may further favor cells whose receptive fields fall in line with the axis given by their orientation tuning (Bosking, Zhang, Schofield, & Fitzpatrick, 1997, but see Martin et al., 2014). In somatosensory cortex, horizontal connections occur between distant cells that respond to sensations at opposing fingertips (Négyessy et al., 2013). This illustrates that horizontal connectivity is not simply determined by receptive field similarity, but more generally seems to favor cells that are activated jointly in common sensorimotor contexts (such as handling an object between two fingers or seeing a continuous line). Conversely, these are contexts that activate well-connected groups of cells.

To provide a concrete illustration, we briefly turn to the visual cortex. V1 cells are known to respond selectively to stimuli with retinal coordinates matching their cortical position (retinotopy) and with a particular angle (orientation tuning). Consider a visual pattern formed by various short line segments. In case these elements are scattered across the visual field, retinotopy implies that a spatially scattered set of cells is activated. Few of these cells will have direct connections, since horizontal connections preferably connect cells with nearby receptive fields. More generally thus, the shortest path between any two responding cells will likely be longer, on average, for such a scattered stimulus than for a more compact stimulus. A similar illustration can be made with respect to orientation. Consider a chain of line segments. In case their orientations are aligned, a spatially neighboring set of cells with similar orientation tuning responds. The cited patterns in V1 connectivity suggest denser connectivity between these cells than between a more heterogenously tuned set of cells that would respond to a chain of heterogenously oriented segments. Again, depending on the distances, this may translate to a difference in the length of paths between the activated cells. A much more complete, idealized, geometrical model of V1 horizontal connection patterns in relation to visual grouping has been provided by Ben-Shahar and Zucker (2004).

In summary, it appears that the cortex in certain situations receives activation patterns that match the existing lateral network structure particularly well, in the sense that they activate closely connected groups of cells. Further, the network structure appears to reflect experience in the sense that commonly coactivated units are better connected. Thus, to show that synchrony reflects the similarity of an incoming spatial pattern to the patterns commonly seen in the past, one primarily needs to show that synchrony reflects the degree to which the current network connects the cells activated by that pattern. The question, therefore, is why, in this case, there is an increase in spike synchrony.

It seems unlikely at first sight that the discussed horizontal network by itself explains the synchrony effect at hand: These connections establish an excitatory (Ts'o et al., 1986; Sincich & Blasdel, 2001), pulsed (chemical) coupling of spiking cells (McGuire, Gilbert, Rivlin, & Wiesel, 1991). In theoretical models of such networks, synchrony, if possible at all, has been found to be unstable in the sense that the phases of different cells spontaneously align and disperse in reaction even to small disturbances (Ernst, Pawelzik, & Geisel, 1998), depending on delay durations, connection strengths, and the network topology (Pérez et al., 2011).

Consequently, at the heart of all existing spiking network models of cortical synchrony are oscillators formed by the interaction of excitatory and inhibitory cells, in which the latter counterbalance the destabilizing effects of excitation (see Wilson & Bower, 1991; Börgers & Kopell, 2005). Here, we argue that the reported findings of cortical synchrony by themselves do not require or imply the presence of excitatory-inhibitory oscillators.

We mentioned that an important aspect of cortical synchrony is its dependence on spatial features of incoming stimulus patterns. The model by Wilson and Bower (1991) is notable for replicating one such effect (Gray et al., 1989) in an excitatory-inhibitory spike-based model. Richer stimulus-dependent synchrony effects are found in a group of models that abstract away from the (spiking) dynamics of biological networks, aiming for an interpretation of cortical synchrony as a binding mechanism. Specifically, these models demonstrate the emergence of synchronous cell assemblies that reflect large-scale, spatial relations within a stimulus, which, for example, leads to an elegant, self-organized solution for visual grouping or image segmentation problems (Li, 1998; Yen & Finkel, 1998; Schillen & König, 1994; Wang, 1995; Wang & Terman, 1995; Finger & König, 2013). These synchronization effects seem to rely on properties of continuously coupled rate or phase models, as they have not yet been demonstrated in pulse-coupled networks of spiking cells. A more exotic approach to achieve synchronous cell assemblies is to reconfigure the network structure for each new stimulus pattern (Yen & Finkel, 1998; Wang, 1995; Yu & Slotine, 2009), though this is biologically motivated in only one case (Wang, 1995) to a certain degree. Finally, it is interesting to note that VanRullen, Delorme, and Thorpe (2001) criticized the whole category of models cited here as implausibly slow for the purpose of stimulus grouping and proposed a fast, feedforward, spike-based grouping model without regard to synchronization.

Here, in an attempt to find a minimal model that has the required computational properties, we provide conditions under which fixed, purely excitatory, chemically coupled spiking networks alone exhibit the required, fast stimulus-dependent synchrony response. One relevant property of such networks is that they can produce temporally ordered spike responses in the presence of noise. Perhaps counterintuitively, noise appears as a beneficial factor in various biological contexts, such as improved neural signal transmission or increased spike time reliability (Ermentrout, Galán, & Urban, 2008; McDonnell & Abbott, 2009). In particular, excitable systems—systems such as spiking neurons, whose response to a sufficient input is stereotypical and followed by a refractory period—are known to engage in periodic oscillations when they receive noise input of some optimal, nonzero amplitude (Lindner, García-Ojalvo, Neiman, & Schimansky-Geier, 2004). This effect (coherence resonance or autonomous stochastic resonance) has been shown in the FitzHugh-Nagumo (Pikovsky & Kurths, 1997), Hodgkin-Huxley (Lee, Neiman, & Kim, 1998), and Morris-Lecar (Balenzuela & García-Ojalvo, 2005) neuron models, and it extends to arrays of multiple, sufficiently coupled spiking neurons—that is, to excitable media (Neiman, Schimansky-Geier, Cornell-Bell, & Moss, 1999). Such networks, when driven by random input, can display regular traveling waves, among other spatially organized activation patterns. We build on these findings and show that this type of emergent, spatial organization of synchronous activity in excitatory networks allows a fast synchrony-based measure of the match between incoming spatial activation patterns and the current network topology.

This letter is structured as follows. We first show that random spike input is transformed to synchronous responses in various excitable, pulse-coupled networks, in particular when stimulating well-connected subpopulations. This basic effect is then used to demonstrate synchrony as a measure of similarity between incoming stimuli and the long-term stimulus history as reflected in current network structures (“familiarity”). Additionally, we show that the effect is compatible with Gray et al.'s (1989) observation of stimulus dependent synchrony and provide some conditions for its occurrence, and we close with a comparison to known cortical dynamics.

2  Results

We study networks of single-compartment Izhikevich spiking neurons (Izhikevich, 2003) with excitatory chemical (pulsed) coupling. A subpopulation receives uncorrelated, stationary, random (Poisson) input spike trains on excitatory and, optionally, inhibitory synapses. As the network responds, the degree of zero time lag spike synchrony of a particular group of cells is measured. This basic setup is shown in Figure 1.

Figure 1:

Basic setup. (a) A stimulus pattern is defined, and uncorrelated random spike trains are sent to the subset of cells specified by the pattern (black cells, b). (b) Activity in the excitatory network is measured in various locations—here, two fixed measurement sites shown in blue. (c) The degree of zero time lag synchrony of the measured locations varies with the stimulus pattern.

Figure 1:

Basic setup. (a) A stimulus pattern is defined, and uncorrelated random spike trains are sent to the subset of cells specified by the pattern (black cells, b). (b) Activity in the excitatory network is measured in various locations—here, two fixed measurement sites shown in blue. (c) The degree of zero time lag synchrony of the measured locations varies with the stimulus pattern.

The model neuron is set to a regime of integrating, class 1 behavior, firing at a much lower rate than the rate of external input pulses (a feature shared by many cortical neurons; Shadlen & Newsome, 1998). Depending on connectivity, networks of such neurons can generate synchronous responses. Figure 2 illustrates this by showing the spike activity in a 4 10 lattice network with eight-nearest-neighbor connectivity, receiving random input in a central, fixed subpopulation of 12 units. The different panels show the network's response given increasing lateral synapse conductances . Coherent spiking already emerges at lower conductances, followed by a regime of coherent chattering, followed at higher conductances by a regime of spreading activity beyond the input-receiving population. The first two regimes are considered in the following. Animations of network activity over time (see the supplementary material M1, M2, M3 or online1) show that a single oscillation cycle is characterized by one or more waves of activation spreading out from one or more (roughly simultaneously appearing) origin points, passing through the input-receiving region until extinction occurs by either collision with another wave or reaching the stimulus boundary. Fewer, broader waves are observed in networks with strong lateral connectivity. In short, the input-receiving region appears to behave as an excitable medium during the presence of a stimulus.

Figure 2:

Model behavior. One second of activity. Top panel: Voltage trace of a single unit. Other panels: Spike activity in a grid network receiving input in a central subpopulation, for increasing lateral synapse conductances from left to right and top to bottom. (The bottom right panel shows a lower upstream input conductance .) Networks shown unrolled row by row along the vertical axis.

Figure 2:

Model behavior. One second of activity. Top panel: Voltage trace of a single unit. Other panels: Spike activity in a grid network receiving input in a central subpopulation, for increasing lateral synapse conductances from left to right and top to bottom. (The bottom right panel shows a lower upstream input conductance .) Networks shown unrolled row by row along the vertical axis.

In general, different stimulus patterns—different choices of input-receiving cells—lead to different degrees of average zero time lag synchrony () among these cells, depending on the connections that exist between them. This is best visualized on a lattice network. Figure 3a shows such a network set in the chattering regime (). For a stimulus that activates cells connected by short paths, these cells fire more synchronously than the more dispersed cells activated by a scattered version of the stimulus. Synchrony differences between different stimuli are discernible after a few spikes. Figure 3b shows synchrony measurements taken over increasingly long time windows (each starting from stimulus onset at ).

Figure 3:

Synchrony reflects the match between input patterns and network structure. (a) Subpopulations with different average path lengths on the given network are driven by random spike inputs (black dots) as the synchrony of the activated cells is measured. (b) Synchrony measured in growing windows. Middle lines denote the median, colored bands the two middle and two outer quartiles, of 50 trials where synchrony was measured in intervals from to the corresponding time on the lower axis. (c) Speed of synchrony divergence between stimuli A and C. The histogram shows in how many trials these conditions could be successfully classified by synchrony after observing a certain average number of spikes. Here, the conditions could be discerned by the synchrony of the first spike wave in most trials.

Figure 3:

Synchrony reflects the match between input patterns and network structure. (a) Subpopulations with different average path lengths on the given network are driven by random spike inputs (black dots) as the synchrony of the activated cells is measured. (b) Synchrony measured in growing windows. Middle lines denote the median, colored bands the two middle and two outer quartiles, of 50 trials where synchrony was measured in intervals from to the corresponding time on the lower axis. (c) Speed of synchrony divergence between stimuli A and C. The histogram shows in how many trials these conditions could be successfully classified by synchrony after observing a certain average number of spikes. Here, the conditions could be discerned by the synchrony of the first spike wave in most trials.

To quantify the information content of such measurements, we compute the length of time for which the network needs to be observed until a naive Bayes classifier correctly infers which stimulus was presented, based on the measured synchrony values. (See section 4 for details.) We report the number of spikes that need to occur, on average over the measured set of cells, until such a classification succeeds (see the histogram panels in Figures 3 and 5). For example, in case of the three stimuli in Figure 3, it is in almost all trials sufficient to observe the degree of synchrony of one or two spike waves. While simple grid networks as shown have the advantage of allowing an easy visualization of topological distances, we note that the same effect also occurs on other network topologies, such as networks where connection probability diminishes with spatial distance and “small world” networks with power-law degree distribution (see the supplemental material).

We showed excitatory spiking networks in which well-connected subpopulations fire synchronously when they receive (noisy) external drive. We have already argued that behavior like this is pivotal in explaining the interplay of cortical response selectivities, learned excitatory horizontal network structures, and experimental observations of synchrony. Further, a connection to stimulus familiarity is implied. Since cortical horizontal connections concentrate between cells that have often been coactivated by previous stimuli, a stimulus that activates well-connected cells is likely similar to (parts of) these past stimuli; they are thus familiar. To make this connection somewhat more explicit, in Figure 4 we turn to networks with a more heterogenous structure, shaped by stimulus patterns. We sample networks with random, local connectivity. Specifically, the probability that two different cells on a integer lattice are connected is set to fall with their Euclidean distance, . The cutoff varies between cell pairs. For each network, a set of random stimulus patterns plays the part of long-term stimulus history; these patterns are taken to have high prior probability (see section 4). In sampling the network, is then relaxed (from to ) between any cells that co-occur in such a pattern (allowing slightly longer links between such cells). These connections are also stronger ( versus 1 elsewhere). As a result, several of the cell pairs coactivated in such patterns end up being directly and strongly connected. In other words, a number of strong excitatory subnetworks have been embedded in the larger network structure (concentrated around certain stimulus configurations), leading to a network that, under suitable external input, can partially behave as an excitable medium. The network structure is then kept fixed, since we are concerned only with network dynamics during short stimulus presentations. A model of the origin of such clustered network structures in terms of an interaction of plasticity rules and axonal delays has been proposed by Izhikevich, Gally, and Edelman (2004).

Figure 4:

Synchrony as a familiarity measure. (a) We sample spatial networks with a distance-dependent connection probability. For each, a set of random stimulus patterns is fixed, and stronger connectivity is allowed to occur between cells coactivated within such patterns. (b) For each network, many new stimulus patterns are sampled and binned by their similarity (degree of overlap) to the set of patterns “imprinted” in the previous step. The box plot shows the degree of zero time lag spike synchrony for 100 samples from each similarity bin, taken across networks and patterns. The final bar shows the subset of the penultimate bin where two or more connections per activated cell have occurred. Insets: Samples of input patterns of different degree of similarity to the imprinted pattern set of one exemplary network.

Figure 4:

Synchrony as a familiarity measure. (a) We sample spatial networks with a distance-dependent connection probability. For each, a set of random stimulus patterns is fixed, and stronger connectivity is allowed to occur between cells coactivated within such patterns. (b) For each network, many new stimulus patterns are sampled and binned by their similarity (degree of overlap) to the set of patterns “imprinted” in the previous step. The box plot shows the degree of zero time lag spike synchrony for 100 samples from each similarity bin, taken across networks and patterns. The final bar shows the subset of the penultimate bin where two or more connections per activated cell have occurred. Insets: Samples of input patterns of different degree of similarity to the imprinted pattern set of one exemplary network.

The stimulus patterns are random and local, in the sense that the probability that a cell is counted as active in a certain pattern falls with the distance (like above) from a randomly chosen center point. After generating the network, additional patterns are sampled to serve as new, incoming stimuli. We define the familiarity of such a new pattern as the fraction of its cells shared with the initial set of patterns that has shaped the network (its similarity to these patterns). This can also be expressed as a prior probability (see section 4). We define 10 intervals (bins) for this familiarity value and, by rejection sampling, generate a number of stimulus patterns in each bin. For generality, the whole sampling procedure of networks and patterns is repeated until 100 examples (different networks with different input patterns) have been drawn in each bin. There are thus three main sources of variance in the synchrony measurements displayed in Figure 4. First, the network sampling procedure is stochastic, meaning that nominally familiar patterns do not necessarily materialize in the form of a well-connected network. To emphasize this factor, the last bar in Figure 4 shows the subset of trials in which at least two connections have been sampled per stimulated cell, that is, the subset of trials in which the network clearly reflects the pattern. Second, variance is introduced by the random initial patterns that define the network structure, which may, for example, overlap. Third, there is some degree of trial-by-trial variability of any single pattern on a single network, as seen in Figure 3.

Despite these factors, an increase in synchrony is apparent as we increase the similarity of incoming stimulus patterns to the patterns embodied in the network structure. This result can be explained in terms of the effect shown before. When an incoming pattern happens to hit an excitable (strongly connected) subnetwork, dynamics as in Figure 3 play out. There, we have shown that the better a group of activated cells is connected by such strong connections, the higher its synchrony is. Here, as in the cortex, these connections are clustered around certain familiar stimulus patterns; hence, stimuli that resemble these patterns produce stronger synchrony. Note that this is not simply a matter of activating cells in the vicinity of strong connections. In Figure 4, the pattern with similarity index 0.56 (bottom row) is located in a relatively strongly connected region but evidently misses part of the relevant subnetwork.

In the cited experimental reports of cortical synchrony, only a few (multi-)electrode recording sites are usually set and stay fixed throughout the various stimulus presentations. Mimicking this scenario, we find that two fixed, small groups of cells fire with increased synchrony if a stimulus activates cells on the direct path between those groups (see Figure 5). This network was generated by the same procedure used in Figure 4, with increased connectivity concentrated in a horizontal line (as would be expected, for example, in a group of similarly orientation-tuned V1 cells often exposed to continuous contours). Lateral conductances are (see panel a) and (see panel b).

Figure 5:

Spike synchrony reflects global stimulus properties. Two stimulus patterns were compared on a fixed network set in the (a) coherent spiking and (b) coherent chattering regime. Top panels show the degree of zero time lag synchrony of two fixed measurement populations. Middle panels show cross-correlograms between the two measured sites for the two stimuli. Bottom insets show the two stimulus patterns (black markers) and fixed measurement sites (blue markers).

Figure 5:

Spike synchrony reflects global stimulus properties. Two stimulus patterns were compared on a fixed network set in the (a) coherent spiking and (b) coherent chattering regime. Top panels show the degree of zero time lag synchrony of two fixed measurement populations. Middle panels show cross-correlograms between the two measured sites for the two stimuli. Bottom insets show the two stimulus patterns (black markers) and fixed measurement sites (blue markers).

The displayed effects are robust to interfering inhibitory input of various temporal structures. Figure 6 shows the same network as Figure 2 under three different settings:

Under uncorrelated inhibition, the stimulus-driven cells receive random inhibitory input spikes in addition to the excitatory drive. As the conductance of inhibitory input synapses increases beyond that of the lateral excitatory connections, synchrony begins to decline gradually. Under pure feedback inhibition (in which each excitatory cell projects to an additional inhibitory interneuron that projects directly back to it), even strong inhibitory conductances have little effect on synchrony. This changes with the addition of lateral inhibition (in which the inhibitory interneuron projects not only back to its driving excitatory cell, but also to the neighbors of that cell). Here again, we see a gradual decline in synchrony as inhibitory conductances increase beyond the strength of the lateral excitatory network. In sum, we find that the presence of moderately strong inhibition of various temporal structures is not sufficient to destabilize the discussed synchronous dynamics but offers a mechanism to dampen them gradually.

Figure 6:

Influence of various types of inhibition. Uncorrelated inhibitory input (top), pairwise recurrent inhibition (middle), recurrent and lateral inhibition (bottom), for inhibitory conductances ranging from zero to four times the excitatory lateral conductance. Left to right: Network visualization with inhibitory connections in red; connection targets marked by thicker line ends. Exemplary spike activity. Synchrony versus inhibitory conductance. The dotted line indicates the degree of synchrony in the unconnected network.

Figure 6:

Influence of various types of inhibition. Uncorrelated inhibitory input (top), pairwise recurrent inhibition (middle), recurrent and lateral inhibition (bottom), for inhibitory conductances ranging from zero to four times the excitatory lateral conductance. Left to right: Network visualization with inhibitory connections in red; connection targets marked by thicker line ends. Exemplary spike activity. Synchrony versus inhibitory conductance. The dotted line indicates the degree of synchrony in the unconnected network.

3  Discussion

We showed examples of noise-driven, pulse-coupled spiking networks that display synchronous responses to random pulse inputs. By directing this input to cells that are closely and strongly connected (or located on a direct strong path between measured network locations), synchrony is increased. In accordance with experiments (Maldonado et al., 2008), synchrony diverges quickly between different stimuli (i.e., within few spikes).

Our motivation for our choice of network model was the properties of cortical long-range horizontal connections. These are distance-limited, excitatory, chemical (pulsed) couplings between spiking cells, with a topology shaped by long-term experience in the sense that commonly coactivated cells are connected more closely and much more strongly (Cossell et al., 2015). But independent, excitatory subnetworks of functionally related cells also exist at smaller scales (Yoshimura, Dantzker, & Callaway, 2005). In fact, since plasticity in cortical networks seems to encourage cluster formation (Izhikevich et al., 2004), such subnetworks may be ubiquitous. For instance, layer 5 cortical microcircuits appear to form a network of small, strongly excitatory subnetworks (Song et al., 2005). The presented dynamics may therefore be found in structures other than the horizontal connections discussed, the more so as these dynamics occur across a broad range of synapse strengths, noise rates, and diversity of network topologies.

The effect appears to be linked to the presence of traveling spike waves within the input-receiving population. This does not imply that the model predicts large wave fronts traveling across the cortex. Given that strong synaptic connections (on which such waves depend) tend to be highly clustered, spreading waves may play out entirely within relatively confined subnetworks, undetectable at the mesoscopic scale. In other words, the proposed excitable dynamics are compatible with situations where no large-scale traveling fronts are observed. Such fronts do, however, occur in certain situations (Sato, Nauhaus, & Carandini, 2012; Takahashi et al., 2015): In visual cortex, they are mostly found during presentations of small, isolated stimuli (Sato et al., 2012; Nauhaus et al., 2009), and their size and number appear to peak directly following a stimulus change, after which they ease off progressively (Shew et al., 2015).

Inhibitory interneurons play no part in creating the presented synchrony effects; conversely, the effect is robust to moderate levels of interfering inhibitory input, which gradually attenuates it. Reports of cortical stimulus-dependent spike synchrony thus do not by themselves imply the presence of excitatory-inhibitory oscillator pairs.

We have isolated excitable dynamics as a common mechanism to explain a number of reports of cortical spike synchrony, including some of the experiments that sparked the binding-by-synchrony debate. Functionally, this leads to a quick, locally computable, spike timing–encoded measure of stimulus familiarity (or, more generally, beyond sensory areas, of the familiarity of a given spatial activation pattern). As noted, horizontal connectivity patterns across the cortex reflect long-term experience in the sense that commonly coactivated units are particularly well connected. In an excitable dynamic regime as presented here, higher spike synchrony is thus expected for input patterns that resemble experience. Hence, for example, synchrony may increase during perceptions of coherent visual stimuli such as connected lines simply because these are often experienced structures in our visual environment, which the network structure reflects. In such early sensory contexts, familiarity may thus express the degree to which a stimulus is structure or noise. Most behaviorally relevant objects have a highly structured appearance (consisting, for example, of Gestalt-like unbroken lines), while other, noisier signals (with fewer spatial correlations) are typically found in less urgently important phenomena, such as background textures. In other words, often experienced, behaviorally relevant stimuli are characterized by high mutual predictability of their different constituent parts, and synchrony appears to signal this (Vinck & Bosman, 2016). The result is a receptive field–like effect on the network level, in which a given group of cells is tuned to certain, well-correlated spatial arrangements of incoming activity and responds with a particular activation signature to these patterns but not to others, constituting a feature extraction step that operates on the level of groups of activated cells.

In more abstract terms, we have proposed that various cortical networks have access to a synchrony-encoded estimate of the prior probability of observing the current input pattern. A first estimate is available directly after the onset of the pattern (since synchrony in the first few spike waves is often already informative), after which precision continually improves. Hence, such a signal could be used early after input onset in a feedforward fashion—for example, to guide attention toward stimuli composed of plausible parts. More generally, estimates of prior probabilities are a prerequisite in Bayesian accounts of perception and learning, but it is unclear how such probabilities are represented neurally (Fiser, Berkes, Orbán, & Lengyel, 2010). We suggest that a spike-based encoding with the presented mechanism allows to rapid generation and transmission of such signals.

4  Methods

4.1  Network Model

The two-dimensional Izhikevich spiking neuron model is a simplification of the Hodgkin-Huxley model of membrane conductances but has a comparable dynamic repertoire (Izhikevich, 2003). Each neuron is given by its membrane potential and recovery variable :
formula
4.1
formula
4.2
Here we consider , . Spikes are discrete events, triggering a reset of the model. Upon crossing the spike detection threshold of  mV, is increased by 12 and is set to mV. The neuron receives input currents from both other cells in the network () and external, upstream sources (). Each of these currents is the sum of a number of individual synaptic currents that evolve according to the nonlinear, chemical synapse model proposed by Destexhe, Mainen, and Sejnowski (1994) (see also Balenzuela & García-Ojalvo, 2005). Intuitively, these synapses cause quickly increasing currents in response to incoming spikes, diminishing somewhat more gradually back to zero if no additional spikes arrive. A more specific description follows, beginning with the excitatory input the neuron receives from its neighbors in the network:
formula
4.3
Here, is the set of cells directly projecting excitatorily to cell (: inhibitorily). The parameter is the maximum conductance of network-internal (lateral) synapses, marked as excitatory by the synaptic reversal potential . Similarly, scales the conductance of inhibitory synapses, with reversal potential . Each synaptic conductance is further modulated by the fraction of open receptors , which varies in accordance with incoming spikes. Specifically, is driven by the concentration of neurotransmitter in the synaptic cleft , which in turn is a pulse of duration after each incoming spike:
formula
4.4
formula
4.5
is parameterized by the rise and decay time constants . The transmitter concentration is given by the product of two heaviside step functions , chosen such that transmitter is present () precisely from time at which a presynaptic spike occurred until time . Upstream connections, responsible for delivering external, random input to the network, can be either excitatory or inhibitory (the parameterization is below). This results in a slightly more complex but essentially similar formulation for the input current . To allow for excitatory and inhibitory input synapses, we introduce a number of additional terms: the conductances of excitatory and inhibitory upstream synapses and , corresponding terms for the fractions of open receptors and , for transmitter concentrations and , and finally for the arrival times of spikes on excitatory and inhibitory input synapses, and .
formula
4.6
formula
4.7
formula
4.8
formula
4.9
formula
4.10
The random excitatory and inhibitory input spikes occur independent of each other and across time and space. Concretely, if neuron is set to receive external input, the number of spike events per ms is Poisson distributed with rates and , respectively. This is realized by independent coin flips with success probabilities and at each numerical integration step, of which are performed per ms.

4.2  Measuring Synchrony

Throughout the letter, we measure the average degree of zero-lag synchrony of a particular subpopulation in the network, following García-Ojalvo, Elowitz, and Strogatz (2004). Spike trains from each neuron are convolved with a causal exponential kernel , yielding an activation trace per neuron. With this, the synchrony of a population during an interval is given by the variance of the mean field of , normalized by the avarage variance of the members of :
formula

Intuitively, if all members of the measured population fire strictly simultaneously, the mean activity of these cells fluctuates just as strongly as each of their individual activities (producing a value of ). If cells fire out of phase, their mean activity is comparably stable, while each individual cell still fluctuates as much as before, leading to a lower, though nonzero, value.

4.3  Measuring Synchrony Differences over Time

To quantify for how long a network needs to be observed until a difference in synchrony between stimuli becomes apparent, we take synchrony measurements in increasingly long time intervals. The earlier a correct classification by stimulus is possible, the more quickly the degrees of synchrony of these stimuli must have diverged. In more technical terms, we perform repeated, independent gaussian naive Baye's classifications on synchrony values measured in growing window increments, as follows.

For each stimulus condition, a copy of the network is driven by a given input pattern, and voltage traces of a subpopulation in this copy are recorded. Each stimulus condition is thus identified with an independent, separately measured population . In each population or condition and each window increment step , the synchrony value is assumed to follow a normal distribution. Parameters and of each such distribution are estimated by the sample mean and sample variance of synchrony values measured in half the available simulation trials. We arrive at an estimated density over possible synchrony values for each measured condition and at each window increment:
formula
Treating this as a likelihood function and assuming (discrete) uniform prior probabilities for the different conditions, the posterior probability that condition is the origin of some newly measured synchrony value at window increment is therefore given by
formula
with iterating all considered stimulus conditions. Finding out how long a given trial needs to be observed until it can be classified thus amounts to finding the window length after which this series of posterior probabilities crosses the decision threshold (0.5 in our case of two stimuli). This duration or, rather, the distribution of such durations across many simulation trials, is reported. More precisely, we report the distribution of number of spikes fired up to that time on average over the measured set of cells.

4.4  Probabilistic Formulation of Familiarity

We have used an intuitive description of pattern familiarity, namely, the fraction of cells in a given pattern that overlaps with the network's input history, which is a set of patterns. This can equivalently be expressed in probabilistic terms. Let the prior probability of occurrence of an input pattern (set of active cells) be given by a joint event of individual activations of its members:
formula
The main simplifying assumption of the familiarity measure is that these individual activation probabilities are binarized, in the sense that a constant, nonzero activation probability is assigned only to cells that appear in patterns found in :
formula
With the choice of normalization , patterns that fall fully within have prior probability , whereas those that miss altogether have probability 0, with intermediate values for partial overlaps. It is in this precise sense that we call synchrony an estimate of the prior probability of an incoming stimulus pattern.

4.5  Code Availability

Annotated source files to reproduce all presented results are found at https://github.com/cknd/synchrony.

Note

Acknowledgments

J.G.O. was supported by the Ministerio de Economia y Competividad and FEDER (Spain, project FIS2015-66503-C3-1-P) and the ICREA Academia programme. E.U. acknowledges support from the Scottish Universities Life Sciences Alliance (SULSA) and HPC-Europa2.

References

Atencio
,
C. A.
, &
Schreiner
,
C. E.
(
2013
).
Auditory cortical local subnetworks are characterized by sharply synchronous activity
.
Journal of Neuroscience
,
33
(
47
),
18503
18514
.
Balenzuela
,
P.
, &
García-Ojalvo
,
J.
(
2005
).
Role of chemical synapses in coupled neurons with noise
.
Physical Review E
,
72
(
2
),
021901
.
Ben-Shahar
,
O.
, &
Zucker
,
S.
(
2004
).
Geometrical computations explain projection patterns of long-range horizontal connections in visual cortex
.
Neural Computation
,
16
(
3
),
445
476
.
Börgers
,
C.
, &
Kopell
,
N.
(
2005
).
Effects of noisy drive on rhythms in networks of excitatory and inhibitory neurons
.
Neural Computation
,
17
(
3
),
557
608
.
Bosking
,
W. H.
,
Zhang
,
Y.
,
Schofield
,
B.
, &
Fitzpatrick
,
D.
(
1997
).
Orientation selectivity and the arrangement of horizontal connections in tree shrew striate cortex
.
Journal of Neuroscience
,
17
(
6
),
2112
2127
.
Brosch
,
M.
,
Budinger
,
E.
, &
Scheich
,
H.
(
2002
).
Stimulus-related gamma oscillations in primate auditory cortex
.
Journal of Neurophysiology
,
87
(
6
),
2715
2725
.
Cossell
,
L.
,
Iacaruso
,
M. F.
,
Muir
,
D. R.
,
Houlton
,
R.
,
Sader
,
E. N.
,
Ko
,
H.
, …
Mrsic-Flogel
,
T. D.
(
2015
).
Functional organization of excitatory synaptic strength in primary visual cortex
.
Nature
,
518
(
7539
),
399
403
.
Destexhe
,
A.
,
Mainen
,
Z. F.
, &
Sejnowski
,
T. J.
(
1994
).
An efficient method for computing synaptic conductances based on a kinetic model of receptor binding
.
Neural Computation
,
6
(
1
),
14
18
.
Ermentrout
,
G. B.
,
Galán
,
R. F.
, &
Urban
,
N. N.
(
2008
).
Reliability, synchrony and noise
.
Trends in Neurosciences
,
31
(
8
),
428
434
.
Ernst
,
U.
,
Pawelzik
,
K.
, &
Geisel
,
T.
(
1998
).
Delay-induced multistable synchronization of biological oscillators
.
Physical Review E
,
57
(
2
),
2150
.
Finger
,
H.
, &
König
,
P.
(
2013
).
Phase synchrony facilitates binding and segmentation of natural images in a coupled neural oscillator network
.
Frontiers in Computational Neuroscience
,
7
,
195
.
Fiser
,
J.
,
Berkes
,
P.
,
Orbán
,
G.
, &
Lengyel
,
M.
(
2010
).
Statistically optimal perception and learning: From behavior to neural representations
.
Trends in Cognitive Sciences
,
14
(
3
),
119
130
.
Galuske
,
R. A.
, &
Singer
,
W.
(
1996
).
The origin and topography of long-range intrinsic projections in cat visual cortex: A developmental study
.
Cerebral Cortex
,
6
(
3
),
417
430
.
García-Ojalvo
,
J.
,
Elowitz
,
M. B.
, &
Strogatz
,
S. H.
(
2004
).
Modeling a synthetic multicellular clock: Repressilators coupled by quorum sensing
.
Proceedings of the National Academy of Sciences of the United States of America
,
101
(
30
),
10955
10960
.
Gray
,
C. M.
(
1999
).
The temporal correlation hypothesis of visual feature integration: Still alive and well
.
Neuron
,
24
(
1
),
31
47
.
Gray
,
C. M.
,
König
,
P.
,
Engel
,
A. K.
, &
Singer
,
W.
(
1989
).
Oscillatory responses in cat visual cortex exhibit inter-columnar synchronization which reflects global stimulus properties
.
Nature
,
338
(
6213
),
334
337
.
Izhikevich
,
E. M.
(
2003
).
Simple model of spiking neurons
.
IEEE Transactions on Neural Networks
,
14
(
6
),
1569
1572
.
Izhikevich
,
E. M.
,
Gally
,
J. A.
, &
Edelman
,
G. M.
(
2004
).
Spike-timing dynamics of neuronal groups
.
Cerebral Cortex
,
14
(
8
),
933
944
.
Jackson
,
A.
,
Gee
,
V. J.
,
Baker
,
S. N.
, &
Lemon
,
R. N.
(
2003
).
Synchrony between neurons with similar muscle fields in monkey motor cortex
.
Neuron
,
38
(
1
),
115
125
.
Kohn
,
A.
, &
Smith
,
M. A.
(
2005
).
Stimulus dependence of neuronal correlation in primary visual cortex of the macaque
.
Journal of Neuroscience
,
25
(
14
),
3661
3673
.
Lee
,
S.
,
Neiman
,
A.
, &
Kim
,
S.
(
1998
).
Coherence resonance in a Hodgkin-Huxley neuron
.
Physical Review E
,
57
(
3
),
3292
.
Li
,
W.
, &
Gilbert
,
C. D.
(
2002
).
Global contour saliency and local colinear interactions
.
Journal of Neurophysiology
,
88
(
5
),
2846
2856
.
Li
,
Z.
(
1998
).
A neural model of contour integration in the primary visual cortex
.
Neural Computation
,
10
(
4
),
903
940
.
Lindner
,
B.
,
García-Ojalvo
,
J.
,
Neiman
,
A.
, &
Schimansky-Geier
,
L.
(
2004
).
Effects of noise in excitable systems
.
Physics Reports
,
392
(
6
),
321
424
.
Livingstone
,
M. S.
(
1996
).
Oscillatory firing and interneuronal correlations in squirrel monkey striate cortex
.
Journal of Neurophysiology
,
75
(
6
),
2467
2485
.
Maldonado
,
P.
,
Babul
,
C.
,
Singer
,
W.
,
Rodriguez
,
E.
,
Berger
,
D.
, &
Grün
,
S.
(
2008
).
Synchronization of neuronal responses in primary visual cortex of monkeys viewing natural images
.
Journal of Neurophysiology
,
100
(
3
),
1523
1532
.
Martin
,
K. A.
,
Roth
,
S.
, &
Rusch
,
E. S.
(
2014
).
Superficial layer pyramidal cells communicate heterogeneously between multiple functional domains of cat primary visual cortex
.
Nature Communications
, 5.
McDonnell
,
M. D.
, &
Abbott
,
D.
(
2009
).
What is stochastic resonance? Definitions, misconceptions, debates, and its relevance to biology
.
PLoS Comput. Biol.
,
5
(
5
),
e1000348
.
McGuire
,
B. A.
,
Gilbert
,
C. D.
,
Rivlin
,
P. K.
, &
Wiesel
,
T. N.
(
1991
).
Targets of horizontal connections in macaque primary visual cortex
.
Journal of Comparative Neurology
,
305
(
3
),
370
392
.
Nauhaus
,
I.
,
Busse
,
L.
,
Carandini
,
M.
, &
Ringach
,
D. L.
(
2009
).
Stimulus contrast modulates functional connectivity in visual cortex
.
Nature Neuroscience
,
12
(
1
),
70
76
.
Neiman
,
A.
,
Schimansky-Geier
,
L.
,
Cornell-Bell
,
A.
, &
Moss
,
F.
(
1999
).
Noise-enhanced phase synchronization in excitable media
.
Physical Review Letters
,
83
(
23
),
4896
.
Négyessy
,
L.
,
Pálfi
,
E.
,
Ashaber
,
M.
,
Palmer
,
C.
,
Jákli
,
B.
,
Friedman
,
R. M.
, …
Roe
,
A. W.
(
2013
).
Intrinsic horizontal connections process global tactile features in the primary somatosensory cortex: Neuroanatomical evidence
.
Journal of Comparative Neurology
,
521
(
12
),
2798
2817
.
Onat
,
S.
,
Jancke
,
D.
, &
König
,
P.
(
2013
).
Cortical long-range interactions embed statistical knowledge of natural sensory input: A voltage-sensitive dye imaging study
.
F1000Research
,
2
.
Pérez
,
T.
,
Garcia
,
G. C.
,
Eguíluz
,
V. M.
,
Vicente
,
R.
,
Pipa
,
G.
, &
Mirasso
,
C.
(
2011
).
Effect of the topology and delayed interactions in neuronal networks synchronization
.
PloS One
,
6
(
5
),
e19900
.
Pikovsky
,
A. S.
, &
Kurths
,
J.
(
1997
).
Coherence resonance in a noise-driven excitable system
.
Physical Review Letters
,
78
(
5
),
775
.
Pipa
,
G.
, &
Munk
,
M. H.
(
2011
).
Higher order spike synchrony in prefrontal cortex during visual memory
.
Frontiers in Computational Neuroscience
,
5
.
Pipa
,
G.
,
Riehle
,
A.
, &
Grün
,
S.
(
2007
).
Validation of task-related excess of spike coincidences based on NeuroXidence
.
Neurocomputing
,
70
(
10
),
2064
2068
.
Read
,
H. L.
,
Winer
,
J. A.
, &
Schreiner
,
C. E.
(
2001
).
Modular organization of intrinsic connections associated with spectral tuning in cat auditory cortex
.
Proceedings of the National Academy of Sciences
,
98
(
14
),
8042
8047
.
Reed
,
J. L.
,
Pouget
,
P.
,
Qi
,
H.
,
Zhou
,
Z.
,
Bernard
,
M. R.
,
Burish
,
M. J.
, …
Kaas
,
J. H.
(
2008
).
Widespread spatial integration in primary somatosensory cortex
.
Proceedings of the National Academy of Sciences
,
105
(
29
),
10233
10237
.
Riehle
,
A.
,
Grün
,
S.
,
Diesmann
,
M.
, &
Aertsen
,
A.
(
1997
).
Spike synchronization and rate modulation differentially involved in motor cortical function
.
Science
,
278
(
5345
),
1950
1953
.
Rioult-Pedotti
,
M.
,
Friedman
,
D.
,
Hess
,
G.
, &
Donoghue
,
J. P.
(
1998
).
Strengthening of horizontal cortical connections following skill learning
.
Nature Neuroscience
,
1
(
3
),
230
234
.
Roelfsema
,
P. R.
,
Engel
,
A. K.
,
König
,
P.
, &
Singer
,
W.
(
1996
).
The role of neuronal synchronization in response selection: A biologically plausible theory of structured representations in the visual cortex
.
Journal of Cognitive Neuroscience
,
8
(
6
),
603
625
.
Rothschild
,
G.
,
Nelken
,
I.
, &
Mizrahi
,
A.
(
2010
).
Functional organization and population dynamics in the mouse primary auditory cortex
.
Nature Neuroscience
,
13
(
3
),
353
360
.
Sato
,
T. K.
,
Nauhaus
,
I.
, &
Carandini
,
M.
(
2012
).
Traveling waves in visual cortex
.
Neuron
,
75
(
2
),
218
229
.
Schillen
,
T. B.
, &
König
,
P.
(
1994
).
Binding by temporal structure in multiple feature domains of an oscillatory neuronal network
.
Biological Cybernetics
,
70
(
5
),
397
405
.
Schmidt
,
K. E.
,
Galuske
,
R. A.
, &
Singer
,
W.
(
1999
).
Matching the modules: Cortical maps and long-range intrinsic connections in visual cortex during development
.
Journal of Neurobiology
,
41
(
1
),
10
17
.
Shadlen
,
M. N.
, &
Movshon
,
J. A.
(
1999
).
Synchrony unbound: A critical evaluation of the temporal binding hypothesis
.
Neuron
,
24
(
1
),
67
77
.
Shadlen
,
M. N.
, &
Newsome
,
W.T.
(
1998
).
The variable discharge of cortical neurons: Implications for connectivity, computation, and information coding
.
Journal of Neuroscience
,
18
(
10
),
3870
3896
.
Shew
,
W. L.
,
Clawson
,
W. P.
,
Pobst
,
J.
,
Karimipanah
,
Y.
,
Wright
,
N. C.
, &
Wessel
,
R.
(
2015
).
Adaptation to sensory input tunes visual cortex to criticality
.
Nature Physics
,
11
(
8
),
659
663
.
Sincich
,
L. C.
, &
Blasdel
,
G. G.
(
2001
).
Oriented axon projections in primary visual cortex of the monkey
.
Journal of Neuroscience
,
21
(
12
),
4416
4426
.
Song
,
S.
,
Sjöström
,
P. J.
,
Reigl
,
M.
,
Nelson
,
S.
, &
Chklovskii
,
D. B.
(
2005
).
Highly nonrandom features of synaptic connectivity in local cortical circuits
.
PLoS Biol.
,
3
(
3
),
e68
.
Stettler
,
D. D.
,
Das
,
A.
,
Bennett
,
J.
, &
Gilbert
,
C. D.
(
2002
).
Lateral connectivity and contextual interactions in macaque primary visual cortex
.
Neuron
,
36
(
4
),
739
750
.
Takahashi
,
K.
,
Kim
,
S.
,
Coleman
,
T. P.
,
Brown
,
K. A.
,
Suminski
,
A. J.
,
Best
,
M. D.
, &
Hatsopoulos
,
N. G.
(
2015
).
Large-scale spatiotemporal spike patterning consistent with wave propagation in motor cortex
.
Nature Communications
, 6.
Ts'o
,
D. Y.
,
Gilbert
,
C. D.
, &
Wiesel
,
T. N.
(
1986
).
Relationships between horizontal interactions and functional architecture in cat striate cortex as revealed by cross-correlation analysis
.
Journal of Neuroscience
,
6
(
4
),
1160
1170
.
VanRullen
,
R.
,
Delorme
,
A.
, &
Thorpe
,
S.
(
2001
).
Feed-forward contour integration in primary visual cortex based on asynchronous spike propagation
.
Neurocomputing
,
38
,
1003
1009
.
Vinck
,
M.
, &
Bosman
,
C. A.
(
2016
).
More gamma more predictions: Gamma-synchronization as a key Mechanism for efficient integration of classical receptive field inputs with surround predictions
.
Frontiers in Systems Neuroscience
,
10
.
Wang
,
D.
(
1995
).
Emergent synchrony in locally coupled neural oscillators
.
IEEE Transactions on Neural Networks
,
6
(
4
),
941
948
.
Wang
,
D.
, &
Terman
,
D.
(
1995
).
Locally excitatory globally inhibitory oscillator networks
.
IEEE Transactions on Neural Networks
,
6
(
1
),
283
286
.
Weiss
,
D. S.
, &
Keller
,
A.
(
1994
).
Specific patterns of intrinsic connections between representation zones in the rat motor cortex
.
Cerebral Cortex
,
4
(
2
),
205
214
.
Wilson
,
M. A.
, &
Bower
,
J. M.
(
1991
).
A computer simulation of oscillatory behavior in primary visual cortex
.
Neural Computation
,
3
(
4
),
498
509
.
Yen
,
S.
, &
Finkel
,
L. H.
(
1998
).
Extraction of perceptually salient contours by striate cortical networks
.
Vision Research
,
38
(
5
),
719
741
.
Yoshimura
,
Y.
,
Dantzker
,
J. L.
, &
Callaway
,
E. M.
(
2005
).
Excitatory cortical neurons form fine-scale functional networks
.
Nature
,
433
(
7028
),
868
873
.
Yu
,
G.
, &
Slotine
,
J.
(
2009
).
Visual grouping by neural oscillator networks
.
IEEE Transactions on Neural Networks
,
20
(
12
),
1871
1884
.