Fast random rewiring and strong connectivity impair subthreshold signal detection in excitable networks

We study dynamical responses in locally paced networks consisting of diffusively coupled excitable units with dynamically adjusted connectivity. It is shown that for weak subthreshold pacing, excessive or strong connectivity impairs the reliable response of a network to the stimulus. Fast random dynamic rewiring of the network also acts detrimentally on signal detection by enforcing a faster relaxation upon the paced unit. Our results indicate that efficient signal processing on excitable complex networks requires tight correspondence between the dynamics of connectivity and the dynamical processes taking place on the network. This, in turn, suggests the existence of ‘function-follows-form’ principles for systems described within this framework.


Introduction
Excitability is an important property of several biological and artificial systems, ranging from neural networks and cardiac tissue to chemical reaction systems and laser optics [1][2][3][4]. Weak perturbations acting upon an excitable steady state typically evoke spiking responses, which after a system-characteristic refractory time fade anew towards the steady state. Excitations can be triggered by stochastic or deterministic inputs, whereby one can observe fascinating phenomena such as, for example, stochastic and coherence resonance in temporal and spatially extended systems [5][6][7][8][9], pattern formation and firing synchronization [10] or self-sustained activity [11] and fast coherent responses [12] in small-world neural networks, to mention but a few. Indeed, especially in neural systems, excitability plays a pivotal role and remains an inspiration for ongoing research efforts, as evidenced by recent focus issues devoted to this subject [13,14]. For example, the excitability of so-called leader neurons has been found crucial for the development of bursts of activity in neural networks [15]. Inspired by preceding experimental observations [16], it was conjectured that the leader neurons form a sub-network that acts as a nucleation area for the bursts upon its initial excitation.
Although the majority of studies has so far considered static networks or has dealt with the network architecture and connection strengths in a probabilistic fashion [17], the focus has recently been shifting towards evolving or adaptive networks [18]. In particular, evolving or dynamically changing interactions in the context of excitable neural networks have been considered previously by Bazhenov et al [19], who examined the role of network dynamics in shaping spike timing reliability. Also, Levina et al [20,21] reported on the existence of self-organized criticality due to dynamical synapses in a pulse-coupled neural network, while coherence resonance due to rewiring has been presented recently by Jiang and Ma [22] in the context of diffusively coupled FitzHugh-Nagumo neurons. Outside the realm of excitable systems, Mondal et al [23] showed that the rapid switching of random links among chaotic maps enhances spatiotemporal regularity of their dynamics. Moreover, synchronization on evolving complex networks has also been studied. Sorrentino and Ott [24] proposed an adaptive strategy based on defining an appropriate potential that each node aims to minimize, and it was shown that a similar technique could be used for adaptive learning of a time-evolving network topology as well [25]. Following these developments, further studies examined the emergence of synchronization on evolving complex networks [26,27] and were subsequently integrated into the broader scope of synchronization on complex networks, as reviewed comprehensively in [28].
Our motivation here is to extend the scope of the above studies and gain an understanding about how the connectivity structure of a network of excitable units could contribute to various aspects of signal processing. We focus on a network's detection of weak periodic signals in the framework of small-world architecture and try to understand how some features of connectivity, such as the fraction of shortcuts between the excitable nodes, determine the ability of a network as a whole to detect the signal that is introduced to one of its units (i.e. the paced unit). We show that connectivity and topology can have counterintuitive effects on signal detection in locally paced networks. Namely, we show that excessively strong connectivity of the paced unit impairs reliable signal detection, and we provide a theoretical argument to explain this effect. We then consider the effect that random dynamic redistribution of connectivity has on the ability of a locally paced network to detect a weak localized stimulus, and we demonstrate that fast random rewiring reduces the response of a network. Our observations suggest that the optimal 3 strategy for weak signal detection in locally paced networks of diffusively coupled excitable units should be tightly linked to activity-dependent rewiring of the connectivity. This, in turn, suggests a practical realization of the 'function-follow-form' principle [29][30][31][32][33] on evolving networks, whereby the structure (or form) of the network defines its function, in this case being the dynamical response to the weak signal.
The paper is structured as follows. In section 2, we describe the employed mathematical model, as well as the interaction networks and other mathematical methods presently in use. The results are presented in section 3. In the last section, we summarize our findings and discuss their potential implications.

The model and methods of analysis
We consider networks of diffusively coupled excitable units, with network topology ranging from practically regular to random. The temporal evolution of the ith unit (time is discretized in our model as an iteration number) is described by the following set of map equations that were developed to capture the rich variety of neuronal dynamics, from regular spiking to selfsustained chaotic bursting [34,35] In equation (1), the variable u i captures the fast component of neural dynamics, whereas v i evolves on a slower time scale. The parameter D represents the strength of diffusive coupling between the given unit and the set of its 'neighbors' (parameterized by the matrix ε i j ), which consists of both the unit's neighbors on the regular lattice and of contributions that come from shortcut links to randomly picked units, constructed as detailed below. For the sake of clarity, we assume that the strength of the coupling D is the same for all pairs of coupled units. Elements of matrix ε i j take the following values: In this work, we consider networks with structures ranging from almost regular, through smallworld, to almost randomized. We start from the ring-like network with regular connectivity, where each unit is coupled to its nearest neighbor on each side of the ring. Then, for each pair of previously unconnected units, the shortcut connection is established with the probability p s . The value of p s determines the topological characteristics of the network (the averaged minimal path length between the units and the clustering coefficient).
In addition to being coupled to the rest of the network, each of the units is subject to temporally and spatially uncorrelated Gaussian noise ξ(t), with intensity σ invariant across the network and not dependent on time, such that The goal of the present study is to understand the effects that dynamic changes in the network's connectivity have on the collective population-wide response to weak local stimulation. To this which is added to the dynamics of variable u. Depending on the values of its parameters, the map model can exhibit different dynamical behaviors [34]. For example, for sufficiently high α (as in figure 1(a), where α = 4.1) an individual map can exhibit self-sustained chaotic bursts that do not require external inputs (σ = 0 and A = 0). To work in the excitable regime, we set α = 1.95. It is important to note that in the excitable regime that we consider here, neither the Gaussian noise (parameterized by σ ) nor the signal (parameterized by f and A) alone is supra-threshold, i.e. the spiking activity of the paced unit occurs only when both the noise and the signal are applied to it, as depicted in figures 1(b)-(e). Even for the joint application of noise and signal, however, the response of the coupled paced unit is quite irregular (the coherence of spiking (see equation (5)) is less than 1) and depends on the coupling strength between the units and on the abundance of shortcut connections (figure 2(c)). Parameter values used throughout this work are β = γ = 0.001 and α = 1.99. Initially, all units are initialized from the fixed point of equation (1), which for α < 2 (and D = 0, σ = 0) is given by u * = −1 In what follows, and unless otherwise specified, we used σ = 9 × 10 −3 , A = 8 × 10 −3 and f = 5 × 10 −4 (in units 1/iteration).
In the setting described above, the ability of a single excitable unit to detect weak periodic stimuli can be conveniently quantified by using the so-called 'coherence of spiking' measure [36]. The coherence of spiking measure C S is defined as a fraction of interspike intervals (ISI) that are within 20% of the stimulus period T : Unless otherwise indicated, the measure given by equation (5) is computed for each excitable unit individually and then the average is taken over all of the units in the network and over a large number (typically 100) of statistically independent realizations.

Results
A brief glance at the network's raster plots (figure 2(a)) suggests that the pattern of collective activity that develops in the network in response to weak pacing does strongly depend on the structure of the network. Both the strength of the coupling (parameterized by D) and the connectivity (parameterized by p s ) can critically affect the collective dynamics. Visual inspection confirms that there might exist an optimal level of coupling strength and fraction of shortcut links that maximize the coherent response to the periodic pacing stimulation. This is in agreement with earlier observations [37,38]. We note, however, that interesting phenomena can be observed when either the coupling is made relatively strong (compare the top and second panels in figure 2(a)) and/or the number of shortcut links is relatively high (compare the top and third panels in figure 2(a)); in either of these cases, naive reasoning would imply that the activity has to be more intense (as a result of stronger coupling and a more connected network), but the actual outcome is exactly the opposite. Thus, in the studied excitable network, the relative contribution of strong coupling and strong connectivity can determine the efficiency with which the weak subthreshold signal is detected. The goal of this work is to elucidate the mechanisms responsible for this counterintuitive effect brought about by the network connectivity and coupling strength.
To quantify the above qualitative observations regarding the effects of coupling and connectivity, we use here the simple measure of output coherence relative to the input signal periodicity. For periodic stimulation, this coherence measure C S is defined as a fraction of the element's interspike intervals that fall within the 20% window of the stimulation period T (see equation (5)). Figure 2(b) shows the dependence of averaged output-to-input coherence on the fraction of shortcut links in the excitable network, for different levels of the coupling strength. Two features are eminent. Firstly, coherence of the network's response falls to zero in the large p s limit (corresponding to a strongly wired network with randomized topology). Secondly, stronger coupling between the units increases the C S only up to a certain value of the coupling strength, and the effect of increased coupling on the coherence depends on the fraction of shortcut links p s ( figure 2(b)). From these observations it follows that there exists a coupling-dependent range of p s values, for which the network is optimally coherent with the weak subthreshold signal that is introduced to the paced unit. This can indeed be observed in figure 2(b) (open squares and closed squares). Since the networks we study are locally paced, it is of interest to examine the signal detection properties of the paced unit as well. In figure 2(c), we show the output-input coherence of the paced unit activity, plotted for different scenarios of the network's connectivity and the coupling strength. The striking difference between the paced unit response (figure 2(c)) and the averaged response of the whole network (figure 2(b)) is evident in the limit of low p s and weak coupling (D = 0.002)-the activity of the paced unit is highly coherent with the input signal, but this is not the case for the activity of other units. Thus, in the limit of weak connectivity and coupling, signal detection is reliable, but signal propagation is not.
Why does the network exhibit low coherence to input when the wiring and/or coupling are excessively strong? In principle, the reduced coherence could be either the outcome of some intrinsic alternation in network dynamics or it could arise because of the reduced ability of the pacer to detect the weak signal. The results shown in figures 2(b) and (c) suggest that the latter cause might hold. To understand the role played by the paced unit, we examine the dynamics of networks for which the paced unit was spared from getting any shortcut connections (it was 7 only connected to its nearest neighbors on the regular lattice). Figure 3(a) (top) shows that this modified network is coherent to input over a much wider range of p s than was the baseline model network. This enhanced response is a direct result of an enhanced response of the paced unit (the bottom panel of figure 3(a)). A more detailed investigation (controlling the level of the paced unit's connectivity to the rest of the network) reveals that the values of C S for the paced element fall sharply as the number of its nearest neighbors (and/or the strength of the coupling to them) is increased (the top left panel of figure 3(c)), and this kind of behavior could be attributed to the smaller number of spikes that the pacer generated in a fixed time interval (the bottom left panel of figure 3(c)). Consistent with this, a change in the signal's amplitude leads to a change in the coherence-connectivity relation of the paced element (right panels of figure 3(c)). However, as the weaker coupling of the paced element increases its coherence with the input signal, the reduced ability to relay the stimulus-related information to other elements results in a decrease of averaged network coherence. This is evident from figure 3(d), where we show the averaged C S of the paced unit's neighbors versus the coherence of that same paced unit. We conclude that the network's coherence in the strong coupling/wiring limit becomes low because of the feedback from the network to the paced element that reduces the latter's ability to detect the signal. This is similar to the dependence of rheobase currents on related parameters in models of gap junction-coupled neuronal ensembles. Thus, in the currently studied excitable system, having non-uniform connectivity, the network's architecture in itself determines its propensity to respond to a subthreshold signal. This complements and extends (to some extent) some of the earlier results (although for a different network model) that showed how signal properties determine the ability of a network to detect the signal [39].
We now offer an explanation for the degrading effect that the strong connectivity and/or strong coupling have on the coherent response to the subthreshold stimulus. In networks of diffusively coupled units, we can rewrite the coupling term in equation (1) as Therefore, the coupling between the units can be looked at as a linear relaxation term (assuming the coupling coefficient is constant). The relaxation time τ R of the coupling is inversely proportional to the strength of coupling; therefore stronger coupling enforces faster relaxation of weak perturbations in the dynamics of the paced unit, thus impairing its response to the subthreshold stimulus ( figure 3(c)). By the same token, higher connectivity (higher p s ) also acts to decrease the coupling-related relaxation time and thus also acts to impair the subthreshold signal detection ( figure 3(c)).
Recently, increasing theoretical [23,24] and experimental [40] effort has been devoted to understand the effects that dynamic rewiring of network connectivity might have on its dynamics. In particular, for the small-world modeling framework, it has been shown that various dynamic properties (such as the propensity to synchronize) can depend on the rate with which shortcut connections are rewired [24]. In another study [23], it was shown that fast dynamic rewiring could bring about the spatiotemporal order into an otherwise chaotic network. We anticipated that in the currently studied excitable network, dynamic rewiring of shortcuts could have strong effects by virtue of its redefining the effective connectivity of the paced unit. Figure 4(a) demonstrates the dependence of C S on the fraction of shortcut links, for different rates of random shortcut rewiring and different strengths of coupling between the units.  The generic effect of fast rewiring is to decrease the coherence of output activity with respect to the input signal. In networks with dynamically rewired shortcuts, the effective shortcut connectivity p * s of each unit is P * S ≈ P S T −1 R , where T R is the period (measured in terms of the number of iterations) with which the shortcut links are rewired [23]. Thus, either fast shortcut rewiring or the abundant presence of fixed shortcuts increases the effective connectivity and, according to our explanation, impairs the subthreshold signal detection by the paced unit. A tangential observation is that the level of p s for which the peak coherence is observed scales inversely with the rewiring time T R (figures 4(b) and (c)).

Discussion
Several recent studies indicate that dynamic rewiring of interactions among units can have important consequences for the organization of activity in networks of excitable units. Here, we have investigated the effects of dynamic connectivity rewiring on the properties of subthreshold signal detection in locally paced networks of diffusively coupled excitable units. Quite counterintuitively, excessively strong connectivity and/or excessively strong coupling was found to impair subthreshold signal detection. This negative effect of connectivity occurred because stronger coupling enforced a faster relaxation on the dynamics of the paced unit, thus reducing its response to the weak stimulus. Consistent with these 'static' effects, fast dynamic rewiring also acted detrimentally on the collective response of the network to the weak stimulus.
The collective response of a locally paced network to the weak stimulus consists of two stages. Firstly, the stimulus has to be locally detected, and secondly, it needs to be quickly and efficiently relayed to the rest of the network. In the networks studied here, fast random rewiring of connectivity impaired the collective reaction of the system to the weak subthreshold stimulus. This raises an interesting question-what should be the rewiring strategy so as not to compromise either one of the stages (detection or transmission of the input) in signal processing? Clearly, the rewiring has to depend in some way on the pattern of ongoing activity, i.e. render the network with adaptive features. This is reminiscent of activitydependent redistribution of connectivity in neural networks and is consistent with the idea of the reciprocal function-form (dynamics-structure) relation. That is, the form (structure) determines the function (dynamics), which in turn affects the form (structure). It remains to be investigated what are the dynamical rules that give rise to optimal signal detection in diffusively coupled networks of excitable units.
The presented results might improve our understanding of the dynamics of weak signal detection and propagation in diffusively coupled excitable networks, such as networks of neurons or astrocytes (a type of glial cells) coupled by diffusive gap junctions [41]. In modeling studies, the connectivity of these networks is usually assumed to be regular (constant number of gap junctions per neuron/astrocyte); however, in reality the distribution of gap junction connectivity is quite broad [42]. How does this broad connectivity, and the emerging heterogeneity of the network's architecture [43], affect signal processing in such biological networks is a question that seems worth exploring in the future by using more specific models.