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Edited by: David Parker, University of Cambridge, United Kingdom

Reviewed by: Rune W. Berg, University of Copenhagen, Denmark; Christopher C. Lapish, Indiana University – Purdue University Indianapolis, United States

*Correspondence: Bogdan Epureanu

†Present Address: Eleni Gourgou, Department of Mechanical Engineering, University of Michigan, Ann Arbor, MI, United States

This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) or licensor are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

Synaptic deficiencies are a known hallmark of neurodegenerative diseases, but the diagnosis of impaired synapses on the cellular level is not an easy task. Nonetheless, changes in the system-level dynamics of neuronal networks with damaged synapses can be detected using techniques that do not require high spatial resolution. This paper investigates how the structure/topology of neuronal networks influences their dynamics when they suffer from synaptic loss. We study different neuronal network structures/topologies by specifying their degree distributions. The modes of the degree distribution can be used to construct networks that consist of rich clubs and resemble small world networks, as well. We define two dynamical metrics to compare the activity of networks with different structures: persistent activity (namely, the self-sustained activity of the network upon removal of the initial stimulus) and quality of activity (namely, percentage of neurons that participate in the persistent activity of the network). Our results show that synaptic loss affects the persistent activity of networks with bimodal degree distributions less than it affects random networks. The robustness of neuronal networks enhances when the distance between the modes of the degree distribution increases, suggesting that the rich clubs of networks with distinct modes keep the whole network active. In addition, a tradeoff is observed between the quality of activity and the persistent activity. For a range of distributions, both of these dynamical metrics are considerably high for networks with bimodal degree distribution compared to random networks. We also propose three different scenarios of synaptic impairment, which may correspond to different pathological or biological conditions. Regardless of the network structure/topology, results demonstrate that synaptic loss has more severe effects on the activity of the network when impairments are correlated with the activity of the neurons.

The network structure/topology of the brain plays an indisputable role in a wide variety of tasks the brain performs (Sporns,

Although the complete human connectome is not available yet, even mapping individual circuits of humans or other animals central nervous system has provided researchers with enormous amount of data to study the network structure/topology of the brain. The non-random structure of the brain networks is a common conclusion of all these studies (Sporns,

The number and the strength of connections that neurons make with each other, create a non-random spatial topology of neuronal networks (Fornito et al.,

EEG (Gaál et al.,

We adopt a neuron model based on the Hodgkin-Huxley formalism. The model features a fast Na^{+} current, a delayed rectifier K^{+} current and a leakage current (Amitai, ^{2}, _{Na} = 24.0 mS/cm^{2}, _{Na} = 55.0 _{K} = −90.0 mV, _{L} = −60.0 mV (Amitai, _{ext} is the external current (measured in μA/cm^{2}) that controls the firing frequency of the neuron. This current is chosen so that the firing frequency of the

The Na^{+} inactivation gating variable ^{+} delayed rectifier activation gating variable ^{+} current is assumed instantaneous, and is modeled by the following function:

The synaptic current from the presynaptic neuron _{s} is the reversal potential for the synaptic current, which is usually considered equal to zero for excitatory synapses. _{ij} is the connectivity weight between the presynaptic neuron _{ij} = 1. When _{ij} = 0, the two neurons are not connected to each other. We define synaptic deficiencies as any values of _{ij} between 0 and 1. _{ij} is the fraction of open receptors, which follows a simple first order kinetic equation given by:
^{−1}ms^{−1} and β = 0.19 ms^{−1} (Destexhe et al., _{j}] is the concentration of neurotransmitters released by the presynaptic neuron _{max} = 1 mM is the maximum concentration of released neurotransmitters by the presynaptic neuron. _{p} = 5 mV and _{p} = 2 mV are constants that determine the steepness and half-activation value of the neurotransmitter release (Hass et al.,

The total synaptic current received by neuron

We use the degree distribution of networks to construct networks with different topological metrics. While small world networks can be constructed systematically (Watts and Strogatz,

The degree of a neuron is defined as the summation of its indegrees (number of its inputs) and outdegrees (number of its outputs). The indegree and outdegree of a neuron can be different since the connections are not necessarily bidirectional. To generate networks with different degree distributions, we chose to concentrate on bimodal distributions, because they are the simplest distributions that are not single modal. Without any constraints, the degree distribution of a purely random network, also known as an Erdös–Rényi model, follows a Poisson distribution (Erdös and Rényi, _{1} and ω_{2} are weights of the modes and _{1} and _{2} are the mean degree values of the modes. Hence, if weights of the modes are equal, then their average must be equal to 20 (for example _{1} = 10 and _{2} = 30 are a valid pair when the weights are equal).

To generate networks with different degree distributions, first, two Poisson distributions with the desired mean values are generated. Then, each of the probability distributions is normalized and weighted as desired so that the integral of the combined bimodal probability distribution function (PDF) is equal to 1. Next, a bin size is chosen, and the combined PDF is integrated over each bin. The result of the integration in each bin shows the number of neurons that must have degrees between limits of that bin. Next, the proper number of neurons is assigned randomly with degrees according to the limits of each bin. Thus, the PDF is converted to a degree distribution for the anticipated network. The next step is to construct a network according to the established degree distribution. For this purpose, a scrambled list is created in which each neuron is repeated at a number of times equal to its degree (Cohen and Havlin,

Implementation of network construction with desired degree (summation of indegree and outdegree) distribution.

Figure

In addition to random networks with a single mode with mean degree value of 20, three bimodal degree distributions with pairs of {15, 25}, {10, 30}, and {5, 35} are used in this study. Even though the two mean values used to build each distribution are imposed, the way that the lists are assembled and then networks are created is random. Therefore, the whole process of network construction is random, which leads to deviations in the number of synapses among different networks. Nevertheless, these deviations are small and negligible as the number of synapses is mainly a function of the network size and the probability of connectivity, which both remain unchanged in this study. However, to minimize the effects of stochasticity arising from the process of network construction on the system's dynamics, 50 realizations of each degree distribution are used to obtain the results.

To quantify synaptic impairment of the network, we define two metrics: the level of impairment and the percentage of impairment. Impairments are implemented in the elements of the adjacency matrix _{ij} is zero, then the two neurons _{ij} is nonzero, then the postsynaptic neuron _{ij} = 0.4. For the same level and percentage of impairment, three possible scenarios of deficiency are used to study the effects of synaptic deficiency in the network.

In the first impairment scenario, synapses are randomly selected and weakened or removed, with equal probability. Conditions in which all neurons in a network can be affected equally may lead to random weakening of synapses. For example, it is possible that aging affects neurons in some regions of the brain with equal likelihood, and weakens the synapses randomly. Such method of synaptic weakening has been also used to model different stages of Alzheimer's disease (Abuhassan et al.,

In the second impairment scenario, neurons that have a higher number of synapses are more likely to be weakened or removed. The hypothesis for this type of defect is based on the significance of intracellular transport. We speculate that such impairment can be linked to pathologies where axonal transport is not functioning properly (De Vos et al.,

In the third impairment scenario, synapses of neurons that are highly active are more likely to be weakened or removed. Thus, this scenario considers the activity of neurons, not just the network topology. This contrasts the second scenario where neurons with more synapses are more likely to suffer from inefficient axonal transport. If such a neuron is not firing frequently, then even an impaired axonal transport might be capable of keeping synapses functional. Nevertheless, if such a neuron is highly active and fires frequently, then defective axonal transport will result in more ineffective synapses compared to a less active neuron which fires less frequently. In fact, synaptic fatigue has already been seen in experimental results even in healthy neurons for high-frequency stimulation (Pozzo-Miller et al.,

Implementation of the three cases of network impairment. Each graph shows how the original bimodal degree (summation of indegree and outdegree) distribution is affected in different impairment scenarios. Each method of impairment affects the original network degree distribution differently, leading to different network dynamics. Level of impairment and percentage of impairment are 1 and 30%, respectively, for all impairment scenarios.

Comparing the degree distribution of the original network (Figure

If a neuronal network is considered as a graph, each neuron is a node and the synaptic connection between each two neurons is an edge. Then, several metrics can be used to describe features of the network based on graph theory (Rubinov and Sporns,

Unlike topological metrics, dynamical metrics depend on the activity of the neurons and their intrinsic properties, such as their excitability. Network synchronization is one of the most widely used dynamical metrics that has been used for the investigation of complex networks (Barrat et al.,

The first dynamical metric we use, is the presence of persistent activity. Persistent activity is a collective behavior of a network that indicates whether the network can sustain its activity for long periods of time, once the initial stimulus is removed. To quantify persistent activity, we declare that a network has persistent activity if even a single neuron has fired at least once during a time window (i.e., the last 200 ms) at the end of a longer period (e.g., a period of 4,000 ms).

All levels of impairment and percentages of impairment for each network need to be examined to determine the sensitivity of the persistent activity to impairments. However, this approach is computationally inefficient. A more effective approach is to estimate the boundary of persistent activity. This boundary is defined as the curve which separates networks without persistent activity (above the curve) from networks with persistent activity (below the curve). To estimate the boundary of persistent activity, for a fixed percentage of impairment, we start from the highest level of impairment and observe the dynamics of the network. If the network activity is not persistent, then the level of impairment is decreased until a level of impairment with persistent activity is found (or the level of impairment reaches zero). Thus, for a fixed percentage of impairment, the boundary of persistent activity shows the maximum level of impairment that allows the network to have persistent activity. After the maximum level of impairment for a fixed percentage of impairment is found, the percentage of impairment is increased, and the networks are re-examined for persistent activity to find the maximum level of impairment for the new fixed percentage of impairment. This process continues until the percentage of impairment is 100%. The percentage of activity and the level of impairment are varied in increments of 10% and 0.1, respectively, to find the boundary of persistent activity.

The quality of the network activity is the second dynamical metric we introduce. We define it as the fraction of all neurons that fire at least once during a time window at the end of a longer period. Thus, this metric is useful for networks at the limit of persistent activity. Higher quality of activity means that more neurons participate in the activity of the network, during the time window used.

All the neurons in the network are initially at rest. At time ^{2}. At

First, we investigated how the topological metrics vary among the different network structures we have studied in this work. The topological metrics shown in Figure

Comparison of clustering coefficients, characteristic path lengths, and rich club coefficients (the three topological metrics introduced in Section Topological Metrics) for three different network structures/topologies, each one with two modes of equal weight and mean degree values in pairs of {15, 25}, {10, 30}, and {5, 35}. The networks with distinct modes can resemble the structure of small world networks (high clustering coefficient) _{normalized} = _{{15, 25}} /_{{20}}.

Figure

Figure

Raster plots for activity of a network with mean degree values of {10, 30}.

Next, we investigated how the network structure influences the dynamical metrics of the network by using four different degree distributions. The first degree distribution has only one mode with mean degree value of 20, which resembles a purely random network. The remaining three distributions have two modes with equal weights and have mean degree values in pairs of {15, 25}, {10, 30}, and {5, 35}. Figure

Boundary of persistent activity and quality of activity for four different network structures/topologies, when targets of impairments are chosen randomly.

For each degree distribution, networks below the boundary have persistent activity (similar to Figure

When the difference between the two mean values of the degree distribution increases, the neurons start to form two clusters with one cluster having higher rich club coefficient than the other (Figure

Figure

Comparison of persistent activity and quality of activity (the two dynamical metrics introduced in Section Dynamical Metrics), for different methods of impairment and network structures. In the first scenario, synapses are randomly impaired. In the second scenario, neurons with more synapses are preferably impaired. In the third scenario, synapses of most highly active neurons are preferably impaired. Details for the scenarios of impairment are provided in Section Impairment Modeling.

All results presented above correspond to bimodal degree distributions with modes that have equal weights. However, different network structures can be constructed by keeping the mean value of one mode constant and varying the weights of each mode (sum of the weights must equal to 1). As the weight of the first mode becomes larger, the mean value of the second mode starts to increase to keep the total mean value constant. The results in Figure

Comparison of persistent activity and quality of activity (the two dynamical metrics introduced in Section Dynamical Metrics), when the weights of degree distribution mode vary. In the first scenario, synapses are randomly impaired. In the second scenario, neurons with more synapses are preferably impaired. In the third scenario, synapses of most highly active neurons are preferably impaired. Details for the scenarios of impairment are provided in Section Impairment Modeling.

Figure

Figure

Our results show the vulnerability of random networks to synaptic loss, compared to networks with bimodal degree distribution. The robustness of networks with bimodal degree distribution can be attributed to their topological metrics, and especially the presence of rich clubs. Our results also show that targeted synaptic loss, which may resemble different pathological or biological conditions, affects the dynamics of networks more, compared to random impairments. Therefore, monitoring the activity of networks has the potential to reveal underlying pathological or biological conditions earlier than symptom-based detection methods.

We have used a model based on the Hodgkin-Huxley formalism that has been previously used successfully to simulate dynamics of neuronal networks (Fink et al., ^{2+} dynamics have been shown to be related with persistent activity of neurons (Fransén et al., ^{2+} dynamics are not captured in this model, previous research has shown that persistent activity can be observed on the network level even when simple integrate-and-fire neurons have been used (Roxin et al., ^{2+} dynamics will not change the conclusions of this work about the influence of network structure on the activity of the network.

As a first step, we used the degree distribution of networks to construct networks with different topological metrics. While completely random connectivity topologies are usually the first choice made when studying the dynamics of neuronal networks,

The rich club coefficients can be used to describe both the persistent activity and the quality of activity for networks with bimodal degree distribution. When networks start to form rich clubs, hubs of highly connected neurons are created, which are also interconnected to each other. During impairments, these hubs can preserve the activity of the whole network. Having a core of highly connected neurons enables such network structures/topologies to maintain self-sustained activity when they experience loss of synapses. Moreover, for such networks, removal of connections between members of rich clubs and neurons outside the rich clubs does not influence the persistent activity significantly because neurons outside the rich clubs are not responsible for maintaining the persistent activity. Neurons in the rich club are also connected to the neurons outside the rich clubs. Hence, they distribute the activity to the whole network. This is the reason why high rich club coefficients coincide with high robustness in our results (Figures

Our results suggest that there is a compromise between quality of activity and persistent activity of neuronal networks. Unlike the integrated persistent activity, the integrated quality of activity shows a nonlinear behavior when the distance between the modes of the degree distribution increases (Figure

The interplay between persistent activity and quality of activity can be considered as an optimization problem. To achieve higher robustness, our results suggest that the number of connections between neurons in rich clubs must increase. However, if the size of the neuronal network and its synapses are constrained to remain the same, then more connections between neurons in rich clubs mean fewer connections between neurons outside rich clubs. Therefore, even though such networks can endure impairments very well and can maintain persistent activity, only few neurons participate in the activity of the whole network and the quality of activity remains low. From this perspective, the network structure/topology can be viewed as a multi-objective optimization problem where the fitness of a network can be determined by both persistent and quality of activity, and the number of neurons and synapses are the constraints. Even though the network optimization can be regarded as an abstract mathematical problem, emergence of certain structures/topologies in networks can also be considered as evolution of these networks in reality (Holland,

In the present study, we have explored also how selective impairment of neurons can affect the dynamics of neuronal networks by investigating targeted weakening of synapses. We have explored how three different scenarios of synapses loss can affect the dynamical features of neuronal networks.

In the first impairment scenario, synapses are impaired randomly, leading to the least impact on the persistent activity for all the network structures/topologies (Figures

In the second impairment scenario, synapses of neurons with larger number of synapses are more likely to be impaired, leading to more damaging effects to the persistent activity of neuronal networks than random impairments of neurons, for all network structures/topologies. The number of synapses a neuron has is a topological feature of a neuronal network. Therefore, if the wiring between neurons in a network is known, this wiring can be used to suggest where the impairments will occur in case of damaged axonal transport. van den Heuvel and Sporns (

In the third impairment scenario, synapses of highly active neurons are more likely to be impaired, leading to the most destructive effect on the persistent activity of all neuronal networks, when compared to the other impairment scenarios. The structure/topology of neuronal networks plays an important role, but the dynamics is important also. The dynamic map of activity in neuronal networks can provide critical information about regions of interest. Other research has similarly suggested that regions of high activity and metabolism can be associated with cellular mechanism involved in Alzheimer's disease (Buckner et al.,

Altogether, we speculate that the transition in the network structure can be used as an indicator of neurodegenerative disease, since the robustness of neuronal networks decreases when they lose their structured topology. Such transition of the brain network toward randomness has already been shown even in normal aging (Knyazev et al.,

EM and BE conceived the idea. EM, EG, VB, and BE designed the experiments. EM created the computational model and performed the simulations. EM, EG, VB, and BE analyzed the data. EM and BE contributed analysis tools. EM and EG wrote the manuscript. All authors reviewed, revised and approved the final manuscript.

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