Abstract
Simple models of excitable dynamics on graphs are an efficient framework for studying the interplay between network topology and dynamics. This topic is of practical relevance to diverse fields, ranging from neuroscience to engineering. Here we analyze how a single excitation propagates through a random network as a function of the excitation threshold, that is, the relative amount of activity in the neighborhood required for the excitation of a node. We observe that two sharp transitions delineate a region of sustained activity. Using analytical considerations and numerical simulation, we show that these transitions originate from the presence of barriers to propagation and the excitation of topological cycles, respectively, and can be predicted from the network topology. Our findings are interpreted in the context of network reverberations and selfsustained activity in neural systems, which is a question of longstanding interest in computational neuroscience.
Similar content being viewed by others
Introduction
The diverse ways in which architectural features of neural networks can facilitate sustained excitable dynamics is a topic of interest in both the theory of complex networks and computational neuroscience. The existence of stable regimes of sustained network activation, for example, is an essential requirement for the representation of functional patterns in complex neural networks, such as the mammalian cerebral cortex. In particular, initial network activations should result in neuronal activation patterns that neither die out too quickly nor rapidly engage the entire network. Without this feature, activation patterns would not be stable, or would lead to a pathological excitation of the whole brain.
A rich and diverse set of investigations has attempted to shed light on the topological prerequisites of selfsustained activity^{1,2,3,4,5,6,7}. One mechanism extensively discussed over the last decade is the phenomenon of reentrant excitations^{4,7,8,9,10}. These reentrant excitations directly couple the cycle content of a graph to properties of selfsustained activity^{4,7,10}. The general phenomenon of cycles serving as dynamical ‘pacemakers’ in the graph is reminiscent of the cores of spiral waves in spatiotemporal pattern formation^{5,8,9}.
Inspired by the importance of selfsustained activity in neuroscience, we here study, using a minimal discrete model of excitable dynamics with a relative excitation threshold, how network topology affects the propagation of excitations through the network. This investigation allows us to develop a mechanistic understanding of the conditions by which a single excitation in a graph amplifies to generate sustained activity. Our contribution with the present paper is two fold:
 1
We investigate how a relative excitation threshold (that is, the minimal fraction of excited neighbors for a node to be excited) affects the usage of structural components (e.g., cycles) in producing reentrant dynamics.
 2
We observe two sharp transitions delineating a region of selfsustained activity. The first transition point corresponds to the onset of excitation propagation between the input node, where a single excitation is injected, and the most distant nodes considered as an output layer; it is similar to the epidemic threshold in epidemic diseases, as observed for example in the SIR model^{11,12,13} (note that in epidemic models, this threshold is in the infection probability, rather than in the relative excitation threshold). The second transition point corresponds to the limit of selfsustained activity and can be related to the occurrence of reentrant excitations.
The approach developed in the present paper provides a simple heuristic to predict, for a given graph and a specific input node, the two transition points observed when varying the relative threshold.
Methods
Dynamical model
We use a threestate cellular automaton model of excitable dynamics on undirected networks. Each node can be in an susceptible/excitable (S), active/excited (E) or refractory (R) state. The model operates on discrete time and employs the following synchronous update rules: For a node i with k_{i} neighbors, the transition from S to E occurs, when at least κk_{i} neighbors are active. The parameter κ thus serves as a relative excitation threshold. In such a scenario, lowdegree nodes are easier to excite (requiring a smaller number of neighboring excitations) than highdegree nodes. Quantitatively, (smallest integer larger than or equal to κk) can be considered as the strength with which a node of degree k acts as a barrier for propagation, by requiring at least n incoming excitations to switch to the excited state.
The model considers only excitatory connections. However, inhibition is implicitly represented in the model due to the automatic transition to a refractory state after excitation. Thus, rather than representing an individual neuron, this model may be thought of as representing a population of coupled excitatory (E) and inhibitory (I) elements as a single node in the network. A node could then for example represent a cortical column consisting of a population of coupled EI neurons with these populations then linked with each other by excitatory connections.
In neuroscience, there is some evidence that a relative threshold criterion is a plausible activation scenario, as neurons can readjust their excitation threshold according to the input^{14}, which typically leads to spike frequency adaptation^{15}, and effectively amounts to a relative input threshold. After a time step in the state E a node enters the state R. The transition from R to S occurs stochastically with the recovery probability p, leading to a geometric distribution of refractory times with an average of 1/p. The model (also investigated before^{16}) does not allow spontaneous transitions from S to E, i.e., compared to previous investigations^{17,18,19}, the probability f of spontaneous excitations is set to zero. Therefore, the stochasticity of the dynamics is entirely due to the stochastic recovery, controlled by the recovery probability p. For p = 1, we have a deterministic model, similar to the one discussed in a previous work^{5}; there, however, a single neighboring excitation was sufficient to trigger transition to E, corresponding to κ → 0.
Details of the numerical experiment
Our numerical experiment starts with a single, randomly chosen input node receiving one excitation, all the nodes being in the susceptible state S. We then observe the propagation of excitations (also termed ‘signal propagation’ in what follows) to an output node, selected at random from the nodes at the largest distance from the input node. We either record the excitations accumulated at this output node during a fixed duration T (typically T = 300 steps, so that the variability of the short transients is averaged out), or observe the absence of propagation reaching the output node. Indeed some of the barriers might not find the required number of active neighbors and fail to propagate the excitation signal. Determinants of successful excitation propagation will thus involve barrier statistics and path multiplicities.
In the present paper, we sample the considered networks in two typical models: random ErdösRenyi (ER) graphs (generated by wiring M edges at random among N nodes) and scalefree BarabásiAlbert (BA) networks (generated with preferential attachment^{20}). For each network, of finite size N, we adopt a layered view (as in a previous investigation^{16}), according to the shortest distance of the nodes to the input node: the first layer contains the neighbors of the input node, the second layer its second neighbors, and the final layer all the possible output nodes, henceforth termed the output layer. By construction, there are no shortcuts between nonadjacent layers. Using this layered view is motivated by the fact that, due to the refractory period, the excitation signal propagates layerwise at low enough κ, moving forward in a coherent way like a front crossing sequentially each layer. This directionality induced by the dynamics itself should not be confused with an intrinsic directionality of the edges: All the networks considered here are undirected.
Some additional technical comments and side remarks for this section and the following sections are provided in the Supplementary Material.
Meanfield analysis of excitation propagation
We here adapt a secondorder meanfield approach from previous work^{16,19} to the present situation estimating the importance of multiple excitations concurring at a given node. The occurrence of such an event at a barrier, i.e. a highdegree node that fails to propagate a single excitation, indeed disrupts our simple prediction topological k^{*} (maximal degree on the easiest path to the output node) of the value 1/κ_{c} of the onset of excitation propagation. By ‘secondorder meanfield approach’, we mean that the computation will use the spatially and statistically average excitation density derived in the meanfield description of the dynamics, but contain a detailed topological analysis of the propagation through a barrier by means of multiple excitations. Only the excitation status of the neighbors of the barrier will be described by the meanfield equations.
Meanfield computation of the excitation probability of a barrier
The concept of barrier simply amounts to consider the number (smallest integer larger than or equal to κk) of excited neighbors required for the excitation of this node of degree k. It describes the strength with which the node acts as a barrier to excitation propagation. In the layered view we have adopted, what matters for a layerwise excitation propagation is not only the strength of a barrier but also the number k^{in} of incoming links from the next upper layer, described through the conditional probability ρ(k^{in}k) given the degree k of the barrier. The probability that a node is a barrier of strength n, but does not act as an obstacle to signal propagation, is thus
where α(nk^{in}) is the probability to have n active nodes among the k^{in} neighbors of the barrier in the next upper layer. This probability can be computed in a meanfield approximation. Considering that a node get excited if its average number kc_{E} of active neighbors is larger than kκ, leading to the meanfield evolution equations (where H is the Heaviside function): c_{E}(t + 1) = c_{S}(t)H[c_{E}(t) − κ], together with c_{S}(t) = 1 − c_{E}(t) − c_{R}(t) and c_{R}(t) = c_{E}(t)/p. This yields a steadystate activity density (provided κ < p/(2p + 1))^{16}. It comes:
Meanfield computation of the probability P_{m} of multiple excitations
The previous computation is based on the assumption that the propagation is consistent with the layered view, moving forward in a coherent way, like a front crossing as a whole each layer. What then matters to get concurrent excitations is the presence of diamond motifs along the paths. We may alternatively consider that the excitations wander along complicated paths. Indeed, we expect that numerous excitation holes (susceptible nodes failing to get excited) exist for κ near κ_{c}, which totally destroys the image of an excitation front propagating layerwise. In this new view, the network around the barrier is well described by an homogeneous activity density , and the excitations could reach a barrier of degree k by any of the k edges, not only those coming from the next upper layer. This amounts to replace k^{in} by k in Eqs (1) and (2). The probability that a node is a barrier of strength n, but does not act as an obstacle to excitation propagation is now simply written (it also accounts for the probability that a node is such a barrier). This latter probability has to be summed over all barrier strengths n ≥ 2, to get the probability P_{m} that multiple concurring excitations at barriers allow the excitation signal to propagate up to the output node:
The related expression 1 − P_{m} will be used to estimate the reliability of our topological prediction k^{*} (largest degree on the easiest path to the output node) of the onset 1/κ_{c} of excitation propagation, which relies on estimating barrier strengths based on single excitations.
Results
Generic properties of the response curve
The outcome of our numerical experiment produces response curves with very similar features, when plotting the excitation level observed at an output node as a function of 1/κ. An example is given in Fig. 1, displaying the three generic features that will be discussed in the following sections: the onset of excitation propagation (point A, value 1/κ_{c}); the limit of selfsustained activity (point B, value 1/κ_{m}) beyond which the excitation signal propagates sequentially through the layers and yields a single record at each output node notwithstanding the duration T of the experiment; and the height of the response curve between these two transition points (level C) which increases with the duration T.
We chose as a control parameter the inverse of the relative threshold κ because 1/κ gives the maximal degree a susceptible node can have to be excited by a single excited neighbor. Any node of degree higher than 1/κ appears as a barrier, that is, a node for which having a single excited neighbor is not sufficient to be excited. With the choice of 1/κ as a control parameter, the transition values can be interpreted in terms of a degree.
The purpose of Fig. 1 is also to show that the transition points of the stochastic model and the deterministic model (i.e. the model with p = 1) coincide. When p < 1, the stochasticity of the recovery makes possible both propagation failure or nonzero output (for the same network and the same input and output nodes) in the range 1/κ_{c} < 1/κ < 1/κ_{m}, as seen in the heat map in Fig. 1. However, the average over a large enough number of runs (30 runs in Fig. 1) yields an average output curve (blue full curve in Fig. 1) displaying the same sharp transitions as the deterministic response curve (red dashed curve in Fig. 1). In fact the deterministic case delimits the possibility space of the stochastic case: All excitation levels that are possible for the deterministic dynamics are in principle achievable in the stochastic case, if the right nodes have recovered at the right moment.
We verified that these various features are generic for any ER graph and any choice of input and output node by extensive simulation, and present in Fig. 2 four more examples obtained with other ER graphs. BA graphs behave in a similar way, and this ensemble of networks will be also considered in the systematic and quantitative study of the response curve features presented in the following sections.
Prediction of the onset of excitation propagation (point A)
All response curves display a transition value 1/κ_{c} for the propagation of a single excitation from the input node to the output layer (point A in Fig. 1). This threshold behavior in the absence of spontaneous excitations is analogous to an epidemic threshold: 1/κ can be roughly interpreted as a transmission probability. It does not depend on the value of p. For a finite network, the transition value 1/κ_{c} is a random variable depending on the realization of the network, the choice of the input node and an output node in the output layer, and the initial conditions.
For 1/κ < 1/κ_{c} (before point A) the relative number of excited neighbors needed to propagate the excitation signal is so large that each possible path from the input node to the output node contains a barrier stopping the propagating excitation signal before it could reach the output node. A single excitation propagating fails to reach the output node if there exists a node of degree k > 1/κ along the path. We thus predict the onset of excitation propagation to arise for a value 1/κ_{c} = k^{*} equal to the smallest (over all linear paths from the input to the output node) of the maximal degree encountered along the path, that is, the largest degree encountered on the easiest path to the output node.
The quantity k^{*} is determined by considering a randomly chosen node in the output layer. A refined prediction for 1/κ_{c} is provided by the largest degree k^{**} encountered on the easiest path to the output layer, that is, by considering all linear paths to any node in the output layer. However, the condition becomes less stringent, if the signal propagation activates redundant paths of the same length, so that more than one excitation may spontaneously arrive at a given node. We thus expect that k^{*} and k^{**} would give only an upper bound on 1/κ_{c}, as seen in Fig. 3.
The quality of our prediction is visualized as a scatter plot comparing the topologically predicted transition value k^{*} and the numerically observed value 1/κ_{c}, Fig. 3. In order to compare the latter value, obtained by a binary search algorithm, with the prediction k^{*}, we round the numerical result to the nearest integer. Exact matches after rounding are then considered a successful prediction. Note that, as we have observed that the transition value does not depend on the value of the recovery probability p, we used p = 1 (the deterministic dynamics) to make the binary search reliable.
We computed the quality of our prediction over a large sample of networks and input nodes as the percentage of cases, where 1/κ_{c} is predicted correctly. In Fig. 4 this prediction quality is shown as a function of the number of edges. As expected, the prediction k^{**} improves the simpler prediction k^{*} in a systematic way. The prediction quality decreases when the graphs become denser. As mentioned above, multiple excitations concurring at a given node, would invalidate our prediction, in allowing this node to be excited even if it were a barrier to the propagation of a single excitation. The probability that a barrier may be passed, because two concurrent excitations reach it, cannot be computed exactly. However we could obtain a meanfield estimate P_{m}, derived in Eq. (3) (see Methods), of the probability of multiple excitations.
As seen on Fig. 1, the critical value 1/κ_{c} does not depend on the value of p. It thus makes sense to look for a purely topological prediction of this value. To check our interpretation that the decrease in quality of our prediction k^{*} is due to concurring excitations alleviating the barrier associated with the node of degree k^{*}, we computed the probability of multiple excitations using a meanfield approach. We chose a value p = 0.5 of the recovery probability in the range where the meanfield approach gives the most robust and reliable approximation of the actual dynamics. The plot of 1 − P_{m} (dashed green curve in Fig. 4) demonstrates that the quality of our prediction k^{*} fails in the way we expected when the link density of the graph increases. Curves of 1 − P_{m} computed for higher values of p similarly match the decrease of the prediction quality, supporting our explanation that the failure of our prediction in dense networks originates in the occurrence of multiple excitations (the more frequent the denser the networks).
The plots presented here have been obtained with networks of N = 80 nodes. The most interesting phenomenon regarding size dependence is the quality reduction for larger BA graphs due to many hubs distorting the propagating front by imposing dynamical directionality^{5} (see Supplementary Material).
Prediction of the limit of sustained activity (point B)
The transition observed in point B is better described when approaching 1/κ_{m} from above. For large values of 1/κ, all nodes have a degree smaller than 1/κ. All nodes are thus able to propagate the activity. Starting from the input node, the excitation propagates layerwise, as a front propagating sequentially across the layers. Each excited layer is followed by a refractory layer, hence no cycling excitations can establish themselves. The only signal amplification comes from the possible branching of paths and an ensuing increase in the number of nodes between one layer and the following one. The single input excitation thus yields a single output excitation per output node, whatever the duration of the observation. Note that the directionality here comes from the dynamics itself (the network is undirected).
When decreasing 1/κ, a jump arises in the output signal at some value 1/κ_{m}. Typically a highdegree node, acting as a barrier, is not excited when the excitation front reaches its layer and leaves a susceptible ‘hole’ in a layer of refractory nodes. Such a hole allows an excitation to travel upward, reaching the hole from an excited neighbor belonging the next layer. The surrounding nodes could be no longer in a refractory state, and complex paths are now available for excitation propagation, possibly including cycling excitations if the network is dense enough to contain suitable topological cycles.
At transition point B we observe a strong amplification of excitations. As was discussed previously^{5,7,10}, such an amplification requires the activation of topological cycles in the graph. We suggest that the transition in 1/κ_{m} can be explained by the appearance of the first active cycle. The amplification observed at point B is sharp. This means that the cycle is traveled several times, or that other cycles can be excited after the first one has stored excitation long enough for some refractory nodes to recover and provide substrate for further cycling excitations.
The analysis of the simulated dynamics in its layered representation for several network realizations shows that as soon as a hole appears in the first layer, other holes rapidly appear in subsequent layers, thus supporting the possibility of numerous cycling excitations, explaining the sharp increase of the output signal (data not shown).
An estimate of the transition point is 1/κ_{m} = k_{max}, where k_{max} is the maximal degree encountered in the network. The degree distribution being layerbiased, it can be expected that with a high probability the first hole appears in the first layer. Accordingly, another prediction is 1/κ_{m} = k_{max,1}, where k_{max,1} is the maximal degree encountered in the first layer.
We compare our prediction k_{max,1} with the numerical value observed for 1/κ_{m} in Fig. 5 (for ER graphs). Again, as we have observed in Figs 1 and 2 that the point B does not depend on the value of the recovery probability p, we used p = 1 (the deterministic case) to determine 1/κ_{m}. Next, we compare our two predictions 1/κ_{m} = k_{max} and 1/κ_{m} = k_{max,1} for ER and BA graphs in Fig. 6 as a function of the number of edges.
For dense networks the prediction k_{max,1} has a 100% quality, meaning that a hole in the first layer is what conditions, directly or indirectly, the onset of a significant amplification of the excitation signal.
In contrast, for sparse graphs, the prediction quality for k_{max} outperforms the one based on k_{max,1}. When 1/κ is small enough for holes to appear in the first layer and be involved in cycling excitation, the signal amplification is already working, due to a hole located in a deeper layer and having a degree k_{max} > k_{max,1}. Indeed in a sparse graph the occurrence of multiple excitations has a low probability P_{m} and holes in deep layers are more frequent than in dense graphs.
Prediction of the height of the response curve (level C)
From the analysis of the topological determinants of selfsustained activity above, it follows that the activity level at point C in Fig. 1 is linked to the presence of cycling excitations, feeding directly or indirectly the output node. The maximal output excitation level would be reached when the output node is periodically excited due to the activity of a set of cycles, with an average period 2 + 1/p prescribed by p. This simple argument describing a saturation of the output node yields a prediction of the maximum excitation level a_{max} at point C depending on p and the length T of the recording:
Figure 7 shows the maximal level reached by the response curve for various values of p as a function of the edge density. As more and more cycles are wired in the network by added edges, the output node excitation level saturates to a pdependent value, whose maximum is reached for p = 1, equal to a_{max} = 100 for T = 300.
The maximal excitation level of the output node as a function of the recovery probability p is shown in Fig. 8, for both ER and BA graphs with N = 80 nodes and a sufficient number of edges (M = 640) to be in the saturation regime (plateaus in Fig. 7). The agreement between the theoretical prediction, Eq. (4) (black line) and the simulation result for p > 0.2 supports our explanation in terms of a complex pattern of active cycles ensuring the recurrent excitation of the output node. For p < 1, the stochasticity of the recovery could allow reexcitation faster than predicted by the average period 2 + 1/p, accounting for the slight excess of the numerical values compared to a_{max}. When p = 1, the dynamics is deterministic and sustained activity originating from robust pacemakers becomes possible^{5}, such as the triangle ESR or the square ESSR. The minimum period of 3 is actually observed for p = 1, and prediction and simulation perfectly match.
The marked discrepancy between the prediction and the simulation at low values of p is expected since the meanfieldlike approximation underlying Eq. (4) is less reliable in the regime of very low excitation densities obtained at low p; here the prediction a_{max} provides only an upper bound on the maximum output excitation level. The kink for small p observed in the actual behavior is due to an inability to sustain the activity due to small graph size combined with many refractory nodes. This finitesize effect is further investigated in Fig. 9, confirming its disappearance for larger graphs. This point emphasizes again the importance of studying small or mediumsized graphs, rather than just the asymptotic limit of infinite graphs, as realworld graphs across all domains of application (from biological to social and technological networks) tend to be comparatively small (with numbers of nodes mostly in the hundreds). In small networks, the topological details like the arrangement of cycles and the barrier structure are of importance for qualitative features of the dynamics, while for infinite graphs these details can be expected to average out.
Discussion
The observed phenomena can be classified as ‘pathdriven’ (for the transition in A; κ = κ_{c}) or ‘cycledriven’ (for the transition in B; κ = κ_{m}). The onset of excitation propagation in A is due to the appearance of the first barrierfree path. The transition in B between simple signal propagation and signal amplification (onset of sustained activity) is due to topological cycles and the possibility of cycling excitation that occurs as soon as some nodes are not excited in the first stage of signal propagation.
Threshold dynamics have been observed in multiple regions of the brain^{21,22}. Computational modeling also demonstrated the power of threshold dynamics in predicting empirical neural activity^{23}. The present model, which uses a global threshold parameter, in line with similar work^{24}, provides a highly simplified representation of threshold mechanisms, which in reality may vary by location as well as in time. However, it is such simplifications that allow the study of the impact of thresholds on network dynamics in the first place, while models that imbue each element with an individual threshold cannot be systematically explored.
Strikingly, the problem of threshold behaviors, which has been investigated in much detail in infectious diseases (e.g., in the SIR model^{12}), has not yet been studied extensively for its impact on global neural dynamics. Only a few studies have contributed to a qualitative understanding of the topological and dynamical prerequisites for selfsustained activity^{4,7,10} (see also the discussion of reentrant excitations in the introduction). While the control of excitation in neural systems may be due to a variety of mechanisms based on cellular properties as well network features^{25}, we here focused intentionally on the latter aspect, notwithstanding the existence or importance of cellular mechanisms.
To our knowledge the deterministic cellular automaton model with a relative excitation threshold considered here has not been investigated before. Previously^{16} a stochastic version of this model was discussed. A similar model^{24} emphasizes that such dynamics can serve as a strategy for exploring the relationship of excitable dynamics and network architecture. It has furthermore been studied^{16} how noise facilitates the propagation of excitations (and in particular can lead to an amplification of excitations in a graph). The gradual change of signal propagation with noise intensity is not related to the sharp, topologically determined transition points discussed in our present investigation. The focus of the previous investigation^{16} was to investigate how the interaction of ‘signal’ excitations (excitations correlated with an input signal) with noise (random excitations) enhances – at intermediate noise levels – the signal recorded at the output nodes. While in the previous work^{16} this stochastic resonance phenomenon could be observed numerically, we are still lacking a deep mechanistic understanding of it. An important prerequisite along this way is to understand how the propagation of a single excitation through the network depends on the network’s topological features and the excitation threshold.
In addition to furthering our understanding of the previous numerical observations^{16}, we believe that the mechanisms outlined here are instrumental to the onset of selfsustained activity in a network. On a more general level, we here advocate the view that, from the perspective of an input node, an excitation ‘sees’ the network as a pattern of paths and barriers.
The layer representation starting from a given input node provides a nodecentered view that a node may have of the network in which it is embedded. This view is relevant in several applications, such as the local probing of a network with no possibility to have an overall and external view, e.g. probing the internet, propagation of signals in neural networks, or social networks in which an individual has only a subjective view of the network.
At intermediate values of 1/κ, the excitation dynamics is sensitive to the hierarchical layer representation of the network. In this sense, we have a processinduced layering, which could also happen in real networks, according to a few input nodes have been specifically selected and evolved to match suitable topological features (or a barrier pattern) for the relevant dynamics.
We restrict ourselves to the analysis of comparatively small networks, as many realworld networks (including the cortical area networks, which serve as a source of inspiration for this investigation) are rather small (a few tens up to a few hundreds of nodes). Moreover, the infinitesize limit would give no clue on the impact, investigated here, of specific topological features on activity propagation. In general, we can expect that the differences between network realizations discussed here (i.e. the variation of 1/κ_{c} and 1/κ_{m} from network realization to network realization) become much smaller with increasing network size.
Our investigation is inspired by the phenomenon of selfsustained activity in biological neural networks. However, we would like to again emphasize that our results are based on a simple model of excitable dynamics with a relative activation threshold operating on an abstract (random) graph. This minimal setup allowed us to understand some generic mechanisms of how network topology influences the propagation of excitations through a graph and how selfsustained activity sets in. However, in spite of the simplicity of the model, we regard these generic results to be valid in more realistic versions of excitable dynamics and, hence, to be of relevance to collective behaviors in the brain.
The feature of our model that certainly requires some scrutiny is the relative threshold. One mechanism potentially leading to such a relative threshold in the modeling of excitable dynamics is spike threshold adaptation that has been described widely for populations of neurons^{26,27,28}. This phenomenon results in a reduction of the firing rate of highly active neural populations, as resulting from a large number of inputs (i.e., for hub regions). While, admittedly, this adaptation mechanism is not identical to the relative threshold mechanism employed in our paper, it points in the same direction: regions with many inputs may become inhibitory barriers in the path of network activation, at least after a transient phase of high activity.
It should further be noted that the emergence of selfsustained activity has also been addressed using different types of models, for instance, where activation and inhibition are taken into account explicitly^{29}.
The transient sustained activity seen in our excitable model is reminiscent of a biological phenomenon termed network reverberation, that is, the temporarily sustained activity induced by a specific stimulation of a neural circuit. The concept is related to that of neural assemblies introduced by Hebb^{30}. One intuitive application of such reverberations may be in dynamic memory circuits, that is, shortterm (working) memory based on dynamic patterns. This is in contrast to longterm memory that may be encoded in the synaptic weight distribution of the network. Indeed, one can see transiently sustained activity in specific cortical regions (e.g., prefrontal and posterior parietal cortex) related to working memory tasks, such as a delayed matchingtosample task. The predominant idea is that reverberations are expressed as dynamic attractors of transiently stable increased activity, particularly due to locally increased synaptic strength^{31}. This idea provides a link between the dynamic patterns encoding shortterm (working) memory and the synaptic weight changes underlying longterm memory. However, there exists an extensive debate on the specific circuitry and parameters underlying the reverberations^{32,33,34}.
Additional Information
How to cite this article: Fretter, C. et al. Topological determinants of selfsustained activity in a simple model of excitable dynamics on graphs. Sci. Rep. 7, 42340; doi: 10.1038/srep42340 (2017).
Publisher's note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
References
Kaiser, M., Görner, M. & Hilgetag, C. Criticality of spreading dynamics in hierarchical cluster networks without inhibition. New Journal of Physics (2007).
Kaiser, M. & Hilgetag, C. C. Optimal hierarchical modular topologies for producing limited sustained activation of neural networks. Front. Neuroinform 4 (2010).
Qian, Y. et al. Diverse selfsustained oscillatory patterns and their mechanisms in excitable smallworld networks. Physical Review E 82, 026107 (2010).
Mcgraw, P. & Menzinger, M. Selfsustaining oscillations in complex networks of excitable elements. Physical Review E 83 (2011).
Garcia, G. C., Lesne, A., Hütt, M. & Hilgetag, C. C. Building blocks of selfsustained activity in a simple deterministic model of excitable neural networks. Frontiers in Computational Neuroscience 6, 50 (2012).
Haimovici, A., Tagliazucchi, E., Balenzuela, P. & Chialvo, D. R. Brain organization into resting state networks emerges at criticality on a model of the human connectome. Phys Rev Lett 110, 178101 (2013).
Garcia, G. C., Lesne, A., Hilgetag, C. C. & Hütt, M.T. Role of long cycles in excitable dynamics on graphs. Physical Review E 90, 052805 (2014).
Qian, Y., Huang, X., Hu, G. & Liao, X. Structure and control of selfsustained target waves in excitable smallworld networks. Physical Review E 81, 036101 (2010).
Liao, X.H. et al. Oscillation sources and wave propagation paths in complex networks consisting of excitable nodes. Frontiers of Physics 6, 124–132 (2011).
Vladimirov, N., Tu, Y. & Traub, R. D. Shortest loops are pacemakers in random networks of electrically coupled axons. Frontiers in Computational Neuroscience 6 (2012).
Anderson, R. M. & May, R. M. Infectious Diseases of Humans (Oxford Univ Press, Oxford, 1991).
Barthelemy, M., Barrat, A., PastorSatorras, R. & Vespignani, A. Dynamical patterns of epidemic outbreaks in complex heterogeneous networks. Journal of Theoretical Biology 235, 275–288 (2005).
Boccaletti, S., Latora, V., Moreno, Y., Chavez, M. & Hwang, D.U. Complex networks: Structure and dynamics. Physics reports 424, 175–308 (2006).
Azouz, R. & Gray, C. M. Dynamic spike threshold reveals a mechanism for synaptic coincidence detection in cortical neurons in vivo . Proceedings of the National Academy of Sciences 97, 8110–8115 (2000).
Benda, J. & Herz, A. V. A universal model for spikefrequency adaptation. Neural Computation 15, 2523–2564 (2003).
Hütt, M.T., Jain, M., Hilgetag, C. C. & Lesne, A. Stochastic resonance in discrete excitable dynamics on graphs. Chaos, Solitons and Fractals 45, 611–618 (2012).
MüllerLinow, M., Marr, C. & Hütt, M. Topology regulates the distribution pattern of excitations in excitable dynamics on graphs. Physical Review E 74, 1–7 (2006).
MüllerLinow, M., Hilgetag, C. C. & Hütt, M.T. Organization of Excitable Dynamics in Hierarchical Biological Networks. PLoS Computational Biology 4, e1000190 (2008).
Hütt, M. & Lesne, A. Interplay between topology and dynamics in excitation patterns on hierarchical graphs. Front. Neuroinform. 3 (2009).
Barabasi, A. & Albert, R. Emergence of scaling in random networks. Science 286, 509 (1999).
Wilent, W. B. & Contreras, D. Stimulusdependent changes in spike threshold enhance feature selectivity in rat barrel cortex neurons. J Neurosci 25, 2983–2991 (2005).
Hu, W. et al. Distinct contributions of na(v)1.6 and na(v)1.2 in action potential initiation and backpropagation. Nat Neurosci 12, 996–1002 (2009).
Gerstner, W. & Naud, R. Neuroscience. how good are neuron models? Science 326, 379–380 (2009).
Haimovici, A., Tagliazucchi, E., Balenzuela, P. & Chialvo, D. R. Brain organization into resting state networks emerges at criticality on a model of the human connectome. Physical Review Letters 110, 178101 (2013).
McCormick, D. A. & Contreras, D. On the cellular and network bases of epileptic seizures. Annu Rev Physiol 63, 815–846 (2001).
Connors, B., Gutnick, M. & Prince, D. Electrophysiological properties of neocortical neurons in vitro . Journal of Neurophysiology 48, 1302–1320 (1982).
Liu, Y.H. & Wang, X.J. Spikefrequency adaptation of a generalized leaky integrateandfire model neuron. Journal of computational neuroscience 10, 25–45 (2001).
Itskov, V., Curto, C., Pastalkova, E. & Buzsáki, G. Cell assembly sequences arising from spike threshold adaptation keep track of time in the hippocampus. The Journal of Neuroscience 31, 2828–2834 (2011).
Vogels, T. P. & Abbott, L. F. Signal propagation and logic gating in networks of integrateandfire neurons. The Journal of neuroscience 25, 10786–10795 (2005).
Hebb, D. O. The Organization of Behavior (Wiley, New York, 1949).
Wang, X.J. Synaptic reverberation underlying mnemonic persistent activity. Trends in Neurosciences 24, 455–463 (2001).
Niktarash, A. H. Discussion on the reverberatory model of shortterm memory: A computational approach. Brain and Cognition 53, 1–8 (2003).
Muresan, R. C. & Savin, C. Resonance or integration? selfsustained dynamics and excitability of neural microcircu its. Journal of Neurophysiology 97, 1911–1930 (2007).
Tegnér, J., Compte, A. & Wang, X.J. The dynamical stability of reverberatory neural circuits. Biological Cybernetics 87, 471–481 (2002).
Acknowledgements
The authors are supported by DFG grants HU 937/71, HI 1286/51 and SFB 936/A1,Z3 as well as TRR 169/A2.
Author information
Authors and Affiliations
Contributions
M.H., A.L. and C.H. conceived the investigation, C.F. performed the numerical simulations. C.F., M.H., A.L. and C.H. analyzed the results. All authors reviewed the manuscript.
Corresponding author
Ethics declarations
Competing interests
The authors declare no competing financial interests.
Supplementary information
Rights and permissions
This work is licensed under a Creative Commons Attribution 4.0 International License. The images or other third party material in this article are included in the article’s Creative Commons license, unless indicated otherwise in the credit line; if the material is not included under the Creative Commons license, users will need to obtain permission from the license holder to reproduce the material. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/
About this article
Cite this article
Fretter, C., Lesne, A., Hilgetag, C. et al. Topological determinants of selfsustained activity in a simple model of excitable dynamics on graphs. Sci Rep 7, 42340 (2017). https://doi.org/10.1038/srep42340
Received:
Accepted:
Published:
DOI: https://doi.org/10.1038/srep42340
This article is cited by

Chromosomal origin of replication coordinates logically distinct types of bacterial genetic regulation
npj Systems Biology and Applications (2020)

Collective sustained oscillations in excitable smallworld networks: the moderate fundamental loop or the minimum Winfree loop?
Nonlinear Dynamics (2020)
Comments
By submitting a comment you agree to abide by our Terms and Community Guidelines. If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate.