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The authors have declared that no competing interests exist.

Conceived and designed the experiments: MZ. Performed the experiments: SM. Analyzed the data: SM. Contributed reagents/materials/analysis tools: SM. Wrote the paper: SM MZ.

Understanding spontaneous transitions between dynamical modes in a network is of significant importance. These transitions may separate pathological and normal functions of the brain. In this paper, we develop a set of measures that, based on spatio-temporal features of network activity, predict autonomous network transitions from asynchronous to synchronous dynamics under various conditions. These metrics quantify spike-timing distributions within a narrow time window as a function of the relative location of the active neurons. We applied these metrics to investigate the properties of these transitions in excitatory-only and excitatory-and-inhibitory networks and elucidate how network topology, noise level, and cellular heterogeneity affect both the reliability and the timeliness of the predictions. The developed measures can be calculated in real time and therefore potentially applied in clinical situations.

The complex dynamics of brain networks underlies information processing as well as various pathologies. Epilepsy [

In this manuscript we take somewhat different approach. We developed a set of measures to study early spatial features of network reorganization upon impending transition into bursting dynamics. Namely, we investigate whether and under what conditions we can identify and later detect early dynamical signs of transitions from synchronous to asynchronous dynamics in highly simplified settings.

In loose terms we assume that the asynchronous mode of activity corresponds to interictal dynamics while synchronous activity corresponds to seizure itself. While this is clearly an oversimplification the goal of this work is to elucidate universal properties of transitions between those two modes of activity.

To make the settings at all relevant to possible clinical applications we the only information we utilize is the relative spatial positions of the neurons and their spiking activity patterns. This could in practice correspond to multiunit information obtained from two or more depth electrodes placed in the brain. We further assume that we have access to this information in the brain region corresponding to localized seizure foci and that transition in this region alone will generate distributed seizure dynamics. We do not tackle the problem of how does synchronous dynamics spread throughout the brain.

We investigate the afore mentioned transitions within ring of excitatory only or excitatory-and-inhibitory integrate-and-fire neuronal networks. This model has been used for more than a century and still is widely used due to its low computational cost, broad range of applications, simplicity along with accuracy [

Even though that the LIF model is one of the simplest models of neuronal dynamics it can reproduce number of biologically observed spatio-temporal patterns depending on the connectivity, synaptic weights, inhibitory feedback, noise and heterogeneity. In 1991 Abeles, showed that if network wires randomly, tight temporal synchrony in order of milliseconds could be easily attained [

Brunel on the other hand investigated the effect of added inhibition into the excitatory oscillators [

Here, we use small-world paradigm to vary network connectivity within the excitatory and inhibitory neurons [

Here, we use the afore mentioned network properties to study early dynamical features of transitions from asynchronous to synchronous network state. We test performance of the developed metric for various network types and network structures. We varied connectivity patterns under two conditions: when the transitions are driven by cellular heterogeneities in the network (i.e. variation of cellular parameters), and 2) when transitions are driven by noise aimed to simulate uncorrelated input from other brain regions to the foci. Those two conditions are to simulate internally and externally driven transitions towards bursting. In sections 1.1 and 1.2 we describe the observed spatio-temporal patterning in the excitatory only and in the excitatory-and-inhibitory networks. Then in section 2 we introduce metrics used to quantify properties of transitions from asynchronous to bursting regimes.

We first characterize the simplified neuronal network dynamics and investigate how distinctive network properties such as the connectivity structure, noise and inhibition can shape its dynamics and influence the properties of transitions between different modes of activity patterns. Here we generally differentiate transitions from/into bursting regime to be driven by noise (modeling uncorrelated input from other parts of the network) and, separately those generated internally by the network, caused by distribution of cell intrinsic frequency.

First we investigated the dynamics of a network consisting of 200 integrate-and-fire excitatory neurons in 1D ring structure and examine its spatio-temporal patterning as a function of noise, external current and its underlying connectivity pattern. The neurons are set to fire spontaneously as they are driven by constant current or random input. The three stimulation types are intended to simulate cellular changes due to the intrinsic neuronal excitation (constant current), input coming from other brain modalities (random input), or both.

It is well established that the dynamics of a neuronal network is highly dependent upon its structure; here we use small-world paradigm to vary network connectivity using excitatory rewiring probability P_{e.} Accordingly, here we show three major classes of network activity patterns can be formed for local, small-world and random topology in _{e} = 0 (_{e} = 0.15), where most connections are local and few of them are rewired to form long distance connectivity. The Small-world regime is known for high clustering and short path lengths and has been shown that the brain possibly shares these connectivity features [_{e} = 0.4, random connections are frequent enough to transform the dynamics into a single synchronized phase. Interestingly in the ISI histogram we see two distant peaks the main peak corresponds to the dominating low-frequency synchronous activity patterns, whereas the small high-frequency peak is due to the asynchronous activity appearing sporadically. Finally, _{e} = 1). Where, we observe stable synchronous bursting with frequency much lower than the small-world regime. _{e} values around the small-world region.

(A-D) Raster plots and ISI histograms associated with deterministic dynamics of networks having P_{e} = 0, 0.15, 0.4 and 1 respectively (blue dots denote timing of neuronal action potential). (E-H) Same as panels (A-D) for noise driven networks (noise frequency = 0.00005). (I) Changes of mean ISIs as a function of rewiring parameter for noise driven identical (I^{e}_{ext} = 1.05, f_{N} = 0.00005 blue line), non-identical (I^{e}_{ext} = 0.95–1.15, f_{N} = 0.00005, green line) and deterministic dynamics for non-identical neurons (I^{e}_{ext} = 0.95–1.15, red line). (J) Changes in mean ISI duration as a function of noise level for an excitatory network with P_{e} = 0.15 for both identical (I^{e}_{ext} = 1.05, blue line), and non-identical neurons (I^{e}_{ext} = 0.95–1.15, green line). Spike timings used for analysis in this figure is provided as the supplemental data in

Changes in the mean ISI values as a function of rewiring parameter are plotted for deterministic dynamics of non-identical neurons and noise-driven dynamics of both identical and non-identical cells (_{e} = 0.0), followed with a significant drop for small-world connectivity regime (P_{e} = 0.15–0.2), and then increase of ISI values in more random network topologies (P_{e}>0.3). This data shows that firing rate is somewhat higher when both heterogeneity and noise present (green line) at P_{e} = 0.15, and it reduced and shifted to P_{e} = 0.2 either with eliminating the noise (red line) or heterogeneity (blue line). Albeit not surprising that the overall frequency increases with addition of noise (additional excitatory input), it is interesting that the frequency changes are more pronounced for random and local network connectivity than the small-world regime. The noise effect on ISIs values for both identical (I^{e}_{ext} = 1.05, blue line) and non-identical neurons (I^{e}_{ext} = 0.95–1.15, green line) are shown in (

Here we will be primarily interested in characterizing transitions between bursting and synchronous activity patterns for different network cellular and network properties. To better illustrate the transitions between the synchronous and asynchronous regime we plotted rasterplot with and example of such transition (

(A, B) An example of raster plot and cumulative network activity pattern for a system composed of 200 excitatory neurons, transitioning from asynchronous to synchronous dynamics. (C) Example of voltage traces of two pairs of neurons ([10, 11],[90,91]) where neurons in each pair are neighbors but the pairs are distant from each other.

Next, we investigated how the various topologies of inhibitory connectivity affect the network’s spatio-temporal patterning. In order to do so, we created two corresponding rings of excitatory and inhibitory cells. The inhibitory neurons send same number of connections as excitatory neurons to other inhibitory and excitatory neurons, but their synapses are weaker than those originating from excitatory neurons. Inhibitory neurons are connected using the same framework as excitatory cells—initially these neurons are connected locally and then rewire part of those local connections based on the inhibitory rewiring parameter (P_{i}), _{e}) fixed and varied the inhibitory rewiring parameter (P_{i} = 0–1). _{e} = 0.15. In case of local inhibition (

(A) Topology of interacting network of excitatory and inhibitory neurons. Here P_{e} = 0.15 and inhibitory connectivity changes from local (P_{I} = 0) to random (P_{I} = 1). (B) Excitatory only neurons with P_{e} = 0.15 when there is no inhibitory feedback. (C) P_{i} = 0, (D) P_{i} = 0.2, the local propagating waves in the asynchronous regime are destroyed. (E) Random inhibitory connections (P_{i} = 1), the firing frequency reduces significantly, while the propagating waves are longer and the synchronous bursting is suppressed. Spike timings used for analysis in this figure is provided as the supplemental data in

The increase of the inhibitory rewiring parameter (P_{i}) causes complex changes to the spatio-temporal firing pattern of the excitatory cells (_{i} values (see _{i}. This is due to rapid spread and equalization of inhibition through out the network.

As we showed above, networks having different properties such as underlying structure, noise and different inhibitory connectivity pattern exhibit distinctive dynamics. In most regimes however we do observe periodic transitions from asynchronous (or less synchronized) to synchronous (or more synchronized) modes of activity. We set out to characterize these different patterns of activity and ultimately elucidate the predictive dynamical features of transitions between these dynamical regimes. In particular we want to investigate under what conditions (if any) these features can be identified sooner rather than later, and thus, reversing the question, can they tell us something about the underlying network properties.

Since the changes in network activity patterns are rapid, we cannot apply measures that are based on long temporal averages, as this would obscure the transition detection. Thus, to characterize the dynamics we developed a set of measures based on assessment of instantaneous changes in adjacent spike-timings of neurons. Based on the observations reported in previous section, the underlying idea of the proposed measures is to analyze, instead of changes in temporal distributions, instantaneous properties of spatial distributions of neuronal activity in given time windows. The major advantage of the developed metrics is that they are simple to compute based on the data that is readily available from recordings and thus can be applied directly to in vivo or clinical measurements. While the exact positions of the recorded cells are clearly unknown, one can ultimately divide the neural populations as coming from the same electrode (cells are nearby) and coming from other electrodes placed at various distances.

The specific question we want to answer is if, and if so, how much before the ultimate synchronous state can we detect changes in spatial network activity patterns. Also, we want to elucidate nature of this transition (e.g. is it a nucleation of locally synchronized groups of neurons)?

Here we divided the spiking data is divided into equal size time-windows with their duration matching the mean ISI observed in the network. Next, we calculate the time difference between closest (temporarily) spikes of every cell that fired within given window and every other cell in the network. These spike timings are then sorted based on the actual spatial distance of neurons (_{D}. We then aim to statistically characterize the properties of this vector as a function of network state, and more importantly near the impending transition into bursting.

(A) To characterize instantaneous spatial patterning in the network, we calculate the minimum time interval between each neuron’s spike in the time-window (blue filled circles) with all other neurons’ spikes and sort these timings based on their spatial distance. The left side of panel A shows the calculation for the a time window that has a asynchronous activity with relatively large and highly variable time intervals; the right side panel depicts calculation for the time window with a synchronous activity and minimal time differences between the spikes. (B) An example of raster plot obtained from noise driven excitatory only network P_{E} = 0.15, noise frequency f = 0.00005 (C) Color plot of consecutive T_{D} calculations; colors indicate the closest timing between spikes of neurons in the given window with all other neurons in the network. (D) Spatial derivative of T_{D} (dT_{D}) in each time window. (E) Mean of T_{D} for consecutive time windows (to which we refer as T_{M}). The dotted line is a cutoff, which we will use to identify the initiation of the bursting dynamics.

_{D} vector for the consecutive time windows (_{D} (dT_{D}), while _{D} for a given time window (T_{M}). We will use the T_{M} to identify the onset and offset of the bursting regime. We do this by setting a threshold value of T_{M} below which we considered that the network dynamics is largely synchronous. While this is to some extend arbitrary the results presented below are (within a range) largely independent of the exact value of the threshold chosen. The dotted line on _{D} are highly dependent on the distance between the cells. The universal property for all network structures (except when P_{e} = 1, see below) is the rapid loss of this distance dependence during the transition. We aim to statistically analyze and characterize properties of these transitions.

The developed metric is quite sensitive to the changes in the network dynamics across various network structures and detects even small variations in the overall observed pattern of activity. An example of such is presented in _{D}, for various connectivity structures of excitatory network (P_{e} = 0; 0.15; 1.0). While the network spatio-temporal patterns are significantly different in the three cases, the metric picks up the bursting regime without difficulty. Moreover the internal structure of the T_{D} vector can shed the light on the intra-burst dynamics of the network. The spatial extend of the changes in T_{D} provides information about the correlation lengths between neuronal activities generated by propagating waves in the network. Thus when all connections are local and the average timing difference between spikes of neurons grows with their actual distance consistent with the long traveling chains of neuronal activities (_{e} = 0.15), we observe much more complex correlation structure with significantly decreased correlation length. The distribution of the local extrema in the T_{D} again corresponds to the shorter chain lengths of activity in the raster plot (_{e} = 1.0), one can still observe changes in timing differences allowing for differentiation of dynamics between less and more synchronous network states. However there is no internal correlation within the given T_{D}. In

Where the left, middle and right column are associated with P_{e} = 0, P_{e} = 0.15, P_{e} = 1 respectively. (A) Raster plots; (B) Spatio-temporal changes of T_{D}; (C) Examples of T_{D} evolution with distance for selected time windows (marked of B); (D) examples of derivative of T_{D} at the same timepoints.

First we set out to investigate the duration of the two (asynchronous and bursting) network regimes. We use the evolution of T_{M} to detect network durations in respective regimes. The threshold defining the onset of the bursting regime is set arbitrarily, however its specific value did not influence significantly the obtained results. We studied duration of the bursting regime for both excitatory only and excitatory-inhibitory networks as a function of topologies of both networks and also as a function of noise level. _{e}) for three types of networks. The first type are the networks composed of identical neurons (same driving excitatory current I_{e} = 1.05; see _{N} = 0.00005). The second network type is not driven by noise, but at the same time its elements are not identical in terms of their driving current and thus their intrinsic firing frequency (I_{e} = 0.95–1.15; note that the mean I_{e} = 1.05). Eventually the third one is driven by noise and its neurons are non-identical in terms of the driving current (f_{N} = 0.00005, I_{e} = 0.95–1.15; mean I_{e} = 1.05, green line). As observed earlier (_{e}. At the same time, for small-world regime, heterogeneity of neurons along with noise (_{e} values (P_{e}>0.3), the neuronal heterogeneity nor/and noise does not change the duration of bursting dynamics significantly.

(A) Fraction of time network adopts synchronous dynamics as a function of rewiring parameter for noise driven identical (I^{e}_{ext} = 1.05 for all neurons f_{N} = 0.00005, blue line), non-identical (I^{e}_{ext} = 0.95–1.15, f_{N} = 0.00005, green line) and deterministic dynamics for network of non-identical cells (I^{e}_{ext} = 0.95–1.15, red line). (B) The effect of the increasing noise level on the dynamics for the excitatory-only network with P_{e} = 0.15 for noise driven identical (I^{e}_{ext} = 1.05 for all neurons, blue line) and non-identical (I^{e}_{ext} = 0.95–1.15, green line).

On the other hand the effect of noise on the dynamics for the excitatory network is illustrated in _{e} = 0.95–1.15 (green line). The fraction of time spent in synchronous dynamics is suppressed with the increased levels of noise, but also depends strongly on cellular heterogeneity.

We analyzed in similar fashion the effect of the inhibitory topology on the spatio-temporal dynamics of the excitatory network (_{i} = 0.95—effectively below spontaneous firing threshold. Thus their firing was driven only by the excitatory network and/or noise. The connectivity of the excitatory networks was kept constant at P_{e} = 0.15 and we varied the inhibitory connectivity (P_{i} = 0.0–1.0). Interestingly the small-world regime of inhibitory connectivity corresponds to the largest fraction of time spent in synchronous dynamics (_{I} = 0.15 (small world topology) and it significantly decreases for random inhibitory network structure. This could indicate that changes in overall inhibitory network structure for example due to axonal sprouting could lead to network more prone to bursting.

The synchronous fraction of dynamics as a function of inhibitory connectivity when excitatory connectivity is in small-world regime (P_{e} = 0.15), for: 1) noise driven identical neurons (f_{N} = 0.00005, I_{e} = 1.05,I_{i} = 0.95; blue line), 2) non-identical neurons (f_{N} = 0.00005, I_{e} = 0.95–1.15, I_{i} = 0.9–1.0; green line), and 3) deterministic dynamics of non-identical neurons (no noise, I_{e} = 0.95–1.15, I_{i} = 0.9–1.0; red line).

The ultimate goal of this study is to characterize network transitions and their predictability from asynchronous activity into the bursting regime. Here we limit the meaning of predictability to identification of first signs of transition to bursting dynamics before the transition itself takes place. Thus, we setout to identify the predictive dynamical features of the transitions as well as their first occurrence relative to the closest transition time, through further analyzing of the T_{D} vector near the transition points. Specifically we utilize measures such as T_{M} (mean value of all T_{D} values), variance from the mean of T_{D} values and variance of dT_{D} (the spatial derivative of T_{D}, see Figs

We want to use measures characterizing properties of T_{D} to detect precursors of the transitions into the bursting dynamics and calculate the lead-time T_{L} (or predictability) to the transition, as a time period before the transitions, during which we can detect significant changes in dynamics, as reported by the developed metrics. First, we measure the values of the above defined metrics in the time windows immediately preceding the onset of the bursts (as defined by the T_{M}). We then calculate the ratios of these values obtained in the consecutive time windows. Thus, we calculate R_{N} = M_{N+1}/M_{N,} where R_{N} denotes the ratio of the (generalized) measure ‘M’ calculated ‘N’ time-windows before the burst onset (N = 0, 1, 2, 3, 4, 5). We then average the ratios over all the realizations of transitions for given network type. If the R_{N} is significantly different from unity we assume that spatial patterning within this time window is persistently and significantly different from that in the prior window. At the same time, the lead-time is defined as the number of time windows prior bursting onset within which the spatio-temporal network pattern undergoes significant change with respect to the one observed in a window before. We defined “predictability” or Lead-time as a number of windows prior to the onset of bursting when the ratio is significantly different from one.

_{D} vector (T_{M}, spatial variance of T_{D} and its spatial derivative, dT_{D} vector, for given time-window) calculated in the times windows N+1 and N before and after the transition (N = 0, 1, 2, 3, 4, 5). _{L}). All three measures used show significant changes before the transition to bursting. The changes in variance of both T_{D} and of dT_{D} show the largest changes before the transition point. However in terms of estimated lead-time T_{M} performs somewhat better (see also Figs

(A-B) the ratio of measures (T_{M}: blue line, Variance Of T_{M}: red line, Variance Of dT_{M}: green line) before and after the onset of the transition into the bursting is shown for P_{i} = 0.2 and 1, respectively; P_{e} = 0.15, f_{N} = 0.00005. Based on these ratios, T_{L} is calculated as a function of the inhibitory connectivity pattern (C). T_{L} peaks for P_{i} = 0.2 and then decreases for more random inhibitory topologies.

(A) P_{e} = 0.0, (B) P_{e} = 0.15 and (C) _{e} _{L} based on the T_{M}, variance of T_{M} and variance of dT_{M} measures respectively.

(A) Excitatory networks (P_{e} = 0.15); (B) excitatory and inhibitory networks (P_{e} = 0.15, P_{i} = 0.0). Solid lines denote simulations in which all neurons receive identical external current (I_{e} = 1.05, I_{i} = 0.95), while dashed-lines are representing simulations with distribution of external currents (I_{e} = 0.95–1.15, I_{i} = 0.9–1.0).

The excitatory network topology is fixed (P_{e} = 0.15) and neurons are identical (I_{e} = 1.05) driven by noise (f_{N} = 0.00005). The connectivity pattern of the inhibitory network is being changed from local (P_{i} = 0) to random (P_{i} = 1). The examples of the ratios of the three metrics at the consecutive time-windows are depicted for P_{i} = 0.2 (_{i} = 1 (_{i} = 0.2.

To better understand the interaction of the excitatory and inhibitory topologies on the lead-time (T_{L}), we explored the effect of inhibition on networks with three excitatory connectivity patterns P_{e} = 0.0, 0.15 and 0.4 having deterministic dynamics (I^{e}_{exc =} 0.95–1.15, I^{i}_{exc} = 0.9–1, _{e}≅P_{i}, as reported by T_{M}.

We next used the measures described above to characterize the effect of noise and variability of neuronal firing frequency on the transition lead-time for both, the excitatory (_{e} = 0.95–1.15 for excitatory cells and I_{i} = 0.9–1.0 for inhibitory cells. Obtained results suggest that predictability is much higher for the networks in which neurons have similar intrinsic firing frequencies, however as expected the lead-time decreases with the increasing noise level.

The Leaky integrate-and-fire (LIF) model was used to simulate network of excitatory and interacting excitatory-inhibitory neurons. The evolution of the voltage across the membrane of neuron ‘j’ is defined as follows:

Where V^{j} and C are the voltage and capacity across j^{th} neuron’s membrane, respectively. The constant α is the leak conductance of the cell that is minimally different for each neuron and chosen from Gaussian distribution (μ = 1, SD = 0. 05). Here I_{ext} is an externally applied current to each cell. Depending on the network model studied (i.e. whether the transitions are due to the noise or to the significant firing frequency mismatch) it can be identical for all of the neurons—external current that excitatory neuron receives: I^{e}_{exc} = 1.05 (the steady state is for most cells just above threshold), external current that inhibitory neuron receives: I^{i}_{exc} = 0.95 (just below the threshold for most cells), or it can be taken from a uniform distribution (I^{e}_{exc} = 1.05, SD = 0. 1 and I^{i}_{exc} = 0.95, SD = 0. 05). After the electrical potential across the cell membrane achieves the threshold set to V_{T} = 1, the cell fires an action potential and its membrane potential is reset to V_{reset}. We set resting-potential ‘E’ and reset-potential ‘V_{reset}‘ equal to zero. Immediately after neuron spikes, the cell enters the refractory period (T_{ref} = 1.5ms).

This synaptic input from presynaptic cell into the postsynaptic cell can be positive or negative depending on its excitatory or inhibitory character of the presynaptic cell and is defined as follows:

Where i and j are presynaptic and postsynaptic neurons, respectively. The ω is the efficacy of connection between presynaptic and postsynaptic neurons; A is the adjacency matrix, H(t)—a Heaviside function, and τ = 1 ms represents the spike duration. We used Euler method with step size h≈0.01ms (estimated from time duration of the spike) to integrate LIF equation for the network.

For networks dynamics incorporating stochastic component, we defined noise as a lighting bolt arriving randomly at each cell with predefined probability. Its arrival at a given site caused the cell to fire instantaneously independent of the membrane voltage, unless the cell was currently spiking or in its refractory time.

The excitatory only network is composed of 200 excitatory neurons forming a 1-D ring structure. The small-world framework was used to vary continuously the network connectivity depending on the rewiring probability [_{e} = 0), to the random connectivity (P_{e} = 1). Every neuron establishes 8 connections to its neighbors (i.e. R = 4).

For interacting excitatory and inhibitory systems we added a corresponding network of 200 inhibitory cells. Thus here the network connectivity pattern is defined by two parameters (P_{e} and P_{i}). Every excitatory cell makes 8 connections to other excitatory and inhibitory cells, every inhibitory cells makes also 8 connections to both other inhibitory and excitatory neurons. The synaptic weight for connections originating from excitatory cells is ω_{e} = 2.2, while that of inhibitory neurons ω_{i} = 0.8.

In order to identify type of the networks’ dynamics (asynchronous versus synchronous) and characterize their transitions, we created a measure based on the relative timing of each neuron’s with respect to other neurons in the system (T_{D}). We divided time of simulation into number of equal length time-windows. The length of the time-window was set to be the average spike frequency in the network. At each time window, the minimum time difference between every neuron’s spike within that window and all other cells is computed (regardless whether the other cells spike within that time-window). If there is more than one spike per neuron in a time window we choose the earliest spike’s time for the given neuron. This calculation is repeated for all the consecutive time windows. These times are then sorted based on the physical distance between neurons and the histogram of the mean times at every distance is resulting in distinct spatial vector T_{D} generated for every time window. The example of T_{D} can is illustrated using a color plot on _{D} values.

We define T_{M} as (spatial) average of T_{D} and use its value to detect temporal location of the transitions into and out of the bursting regime. To characterize the properties of the T_{D} vector near the transition point we calculate its spatial derivative dT_{D}. Finally we calculate dT_{M}, which is the average value of dT_{D} and also variance of T_{D} and dT_{D}.

We want to use measures characterizing properties of T_{D} to detect precursors of the transitions into the bursting dynamics. We calculate the values of the above defined measures in the time windows immediately preceding the onset of the bursts (as defined by the T_{M}). We then calculate the ratios of the measures progressing forward in time. Therefore we calculate R_{N} = M_{N+1}/M_{N}), the ratio of the (generalized) measure ‘M’ calculated ‘N’ time-windows before the burst onset (N = 0, 1, 2, 3, 4, 5). If the R_{N} is significantly different from unity we assume that spatial patterning within this time window is persistently and significantly different from that in the prior window. The lead-time is defined as the number of time windows prior bursting onset within which the spatio-temporal network pattern undergoes significant change with respect to the one observed in a window before. This lead-time is then averaged over many realizations of bursting transitions.

In this study we investigated predictability of network transitions into bursting regime as a function of network structure, cell variability and noise. Initially, we characterized the dynamics for different parameter sets and then we used the developed measures to predict transitions to synchronous activity using spike timings. The networks, as predicted exhibit different types of dynamics, ranging from propagating waves of activity, through coexistence of two phases with short waves of activity and bursting, and finally synchronous dynamics. Addition of inhibition to network shortens the propagating waves, with the transition to bursting suppressed for random inhibitory topologies.

Over the last few decades amount of research is associated with finding robust measures that can detect synchrony [

The introduced measures centered on analysis of relative spike timings of all firing cells within a given time window. The metrics characterized instantaneous spatial correlations between the cells as a function of their physical distance. The systematic changes in the introduced measures in the time windows preceding the bursting onset, were able to predict transition into bursting within few time windows of its onset. It is important to note however that the approach taken does not allow estimating the false positives (i.e. when observed change does not lead to bursting transition), resolving these changes from the ones leading to bursting onset is a subject of ongoing research. The performance of the metrics depended on network topology, noise level and distribution of cellular firing rates. The constructed metrics provide an alternate approach toward gaining an insight on transitions between asynchronous and bursting dynamics. Their advantages are that they can be computed rapidly and thus applied online in clinical use.

Excitatory only network, P_{e} = 0, neurons receive no noise, and their external current is taken from a uniform distribution of I^{e}ext = 0.95–1.15.

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Excitatory only network, P_{e} = 0.15, neurons receive no noise, and their external current is taken from a uniform distribution of I^{e}ext = 0.95–1.15.

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Excitatory only network, P_{e} = 0.4 neurons receive no noise, and their external current is taken from a uniform distribution of I^{e}ext = 0.95–1.15.

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Excitatory only network, P_{e} = 1, neurons receive no noise, and their external current is taken from a uniform distribution of I^{e}ext = 0.95–1.15.

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Excitatory only network, P_{e} = 0, with noise frequency of fN = 0.00005 and all neurons receive identical external current I^{e}ext = 1.05.

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Excitatory only network, P_{e} = 0.15 with noise frequency of fN = 0.00005 and all neurons receive identical external current I^{e}ext = 1.05.

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Excitatory only network, P_{e} = 0.4, with noise frequency of fN = 0.00005 and all neurons receive identical external current I^{e}ext = 1.05.

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Excitatory only network, P_{e} = 1, with noise frequency of fN = 0.00005 and all neurons receive identical external current I^{e}ext = 1.05.

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Excitatory only network, P_{e} = 0.15, neurons receive no noise, and their external current is taken from a uniform distribution of I^{e}ext = 0.95–1.15.

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Excitatory neurons in the excitatory-inhibitory network, where P_{e} = 0.15 and P_{i} = 0, neurons receive no noise, and their external current is taken from a uniform distribution of I^{i}ext = 0.9–1, I^{e}ext = 0.95–1.15.

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Excitatory neurons in the excitatory-inhibitory network, where P_{e} = 0.15 and P_{i} = 0.2, neurons receive no noise, and their external current is taken from a uniform distribution of I^{i}ext = 0.9–1, I^{e}ext = 0.95–1.15.

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Excitatory neurons in the excitatory-inhibitory network, where P_{e} = 0.15 and P_{i} = 1, neurons receive no noise, and their external current is taken from a uniform distribution of I^{i}ext = 0.9–1, I^{e}ext = 0.95–1.15.

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This work was supported by NSF CMMI grant no: 1029388.

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