Bursting Dynamics of Spiking Neural Network Induced by Active Extracellular Medium

Abstract

We propose a mathematical model of a spiking neural network (SNN) that interacts with an active extracellular field formed by the brain extracellular matrix (ECM). The SNN exhibits irregular spiking dynamics induced by a constant noise drive. Following neurobiological facts, neuronal firing leads to the production of the ECM that occupies the extracellular space. In turn, active components of the ECM can modulate neuronal signaling and synaptic transmission, for example, through the effect of so-called synaptic scaling. By simulating the model, we discovered that the ECM-mediated regulation of neuronal activity promotes spike grouping into quasi-synchronous population discharges called population bursts. We investigated how model parameters, particularly the strengths of ECM influence on synaptic transmission, may facilitate SNN bursting and increase the degree of neuronal population synchrony.

1. Introduction

Synchronous neuronal population dynamics attracts great attention in modern neuroscience and neurodynamics in particular. The reason for this is that synchronization underlies many cognitive functions [1], including sleep [2], memory [3], attention [4,5], as well as pathological manifestations [6], for example, in epilepsy [2,7]. Neuronal networks in dissociated neuronal cultures exhibit quasi-synchronous dynamics in the form of so-called population burst discharges [8,9,10,11,12,13]. Such bursts represent high-frequency sequences of spikes elicited by almost all network neurons within a certain time window. The population bursts have different activation profiles and may be used to encode different network dynamical states [13].

Among the known mechanisms of neuronal synchronization, the following groups can be distinguished: properties of neurons [14], properties of neural networks [15,16,17,18], neuromodulation [19] mediated by neuromodulators of various natures. All the presented groups have modulation times on the order of milliseconds, with the exception of neuromodulation mediated by glial cells, which occurs on timescales of seconds [20,21,22,23,24,25,26,27]. In this regard, the search for regulatory mechanisms on longer timescales, comparable to the timescales of brain rhythms, is an urgent task.

More recently, brain extracellular matrix molecules (ECM) have been shown to modulate the efficiency of synaptic transmission and neuronal excitability. It has been suggested that this brain structure plays a key role in the homeostatic regulation of neuronal activity on relatively large timescales [28,29]. ECM-induced homeostatic plasticity prevents pathological changes or even cell death due to the hypo or hyperexcitation of neurons. One of the mechanisms is synaptic scaling, which helps maintaining the level of excitation of neurons within a certain range in response to various changes in afferent inputs [30,31]. This is associated with a change in the concentration of ECM receptors (integrins), leading to alterations in the expression of $\alpha$-amino-3-hydroxy-5-methyl-4-isoxazolepropionic acid (AMPA) receptors, ultimately affecting the formation of postsynaptic currents [28]. Another regulatory mechanism is based on changing the threshold of neuron excitability through the action of heparan sulfate proteoglycans on L-type calcium channels (L-VDCC) [32]. Furthermore, the feedback associated with the regulation of ECM concentration is realized not only through the secretion of ECM into the extracellular space, but also through the activity of proteases (such as tissue plasminogen activator, plasmin, matrix metalloproteinases 2 and 9, agrecanases 1 and 2, neuropsin, and neurotrypsin) that are released pre and postsynaptically, and cleave ECM. Several experimental studies on interneurons have shown that the participation of neuronal L-VDCCs regulates the neuron excitability threshold in the context of ECM–neuronal interaction. Removal of ECM (for example, by proteases) in experiments contributes to the excitation of interneurons.

One of the first mathematical models describing the homeostatic regulation of neuronal activity by ECM was firstly proposed by Kazantsev et al. [29]. The model was built based on kinetic activation functions to describe the ECM activity, utilizing the formalism of the Hodgkin–Huxley model [33]. Action potentials or spikes in presynaptic neurons trigger the release of neurotransmitters from the presynaptic terminal. These neurotransmitters then reach the postsynaptic neuron, where they activate receptors on the postsynaptic membrane, leading to the formation of a postsynaptic current. Each spike generated by a neuron causes an increase in the average neuronal activity. As the average neuronal activity increases, the concentration of molecules in the extracellular matrix of the brain also increases. This increase in ECM concentration leads to an increase in synaptic weights, which in turn further increases the average neuronal activity. Once the threshold value of the average neuronal activity is reached, the concentration of proteases that cleave ECM increases to ensure homeostatic regulation, ultimately resulting in a decrease in neuronal activity.

It has been experimentally shown that the extracellular matrix of the brain “may contain memory traces of local neural network activity” [28] and participates in the modulation of various types of memory [34], as well as potentially contributing to the formation of epileptogenesis [35,36,37]. Burst activity is known to underlie both information processing [38] and storage [39,40,41], including in memory traces [42], as well as in neuropathologies such as epilepsy [43]. Recent experiments on cell cultures of neurons have demonstrated that the extracellular matrix of the brain enhances connections in the mature brain and accelerates the development of neural networks [44], regulating neuronal activity, including burst activity [44,45]. This effect of the extracellular matrix of the brain is also significant from an applied perspective for the accelerated formation of mature neural networks for drug testing and therapeutic methods [45].

In this paper, we consider a spiking neural network interacting with an active extracellular field modeling the effect of the brain extracellular matrix. We demonstrate that ECM modulation of neuronal activity can lead to quasi-synchronous dynamics in the form of population bursts. We also analyze how the degree of network synchrony depends on the strength of the ECM mediated feedback.

The paper is organized as follows. In Section 2, we describe the mathematical model and methods used in this study. In Section 3, we present the main findings of our investigation of the model with and without the influence of the extracellular matrix on neuronal activity due to the effect of synaptic scaling. We demonstrate that this influence can lead to the formation of burst activity. In Section 4, we discuss the results obtained and propose possible directions for further research. Finally, in Section 5, we provide a summary of the study’s results.

2. The Model

For illustrative purposes, we arrange the neurons of our network model on a two-dimensional space. Each neuron is described by the Izhikevich model, which is widely used in network modeling due to its computational efficiency and functionality [46]:

$$\left\{\begin{array}{c}{C}_m{\displaystyle \frac{d{V}_i}{dt}}=0.04V_i^2+5V_i+140-U_i+I_{ex{t}_i}+I_{sy{n}_i},\hfill \\ {\displaystyle \frac{d{U}_i}{dt}}=a(b{V}_i-U_i),\hfill \\ \mathrm{if} V_i\ge 30\mathrm{mV},\mathrm{then}\hfill \\ V_i=c,\hfill \\ U_i=U_i+d,\hfill \end{array}\right.$$

(1)

where the parameters $a,b,c,d$ determine the dynamics of membrane potential $V_i$, $U_i$ is an auxiliary variable describing the process of activation and deactivation of potassium and sodium membrane channels, respectively, and $I_{ex{t}_i}$ is the external current whose values at the initial moment of time are randomly distributed from 0 to $I_{ext}^{max}$. When the membrane potential $V_i$ reaches 30 mV, an action potential (spike) is formed and the values of the variables change.

We set all neurons to their excitable mode when single spikes are generated in response to an external stimulation current. We fixed the following parameter values: $a=0.02$, $b=0.5, c=-40\mathrm{mV}, d=100, k=0.5, C_m=50, I_{ext}^{max}=40$. Term $I_{sy{n}_i}$ represents the sum of synaptic currents from all M presynaptic neuron:

$$I_{sy{n}_i}=\sum _{j=1}^M{y}_{i,j}{w}_{i,j},$$

(2)

where $I_{sy{n}_i}$ is the summation of all synaptic currents of the postsynaptic neuron, the parameter $w_{i,j}$ denotes the weights of glutamatergic and gamma-aminobutyric acid (GABAergic) synapses between pre (i) and postsynaptic (j) neurons, the parameter M describes the number of presynaptic neurons that have nonzero connection with the j-th neuron. For excitatory and inhibitory synapses, the weights are positive and negative, respectively. Variables $y_{i,j}$ denote the output signal (synaptic neurotransmitter) from the i-th neuron to the j-th neuron involved in the generation of $I_{sy{n}_i}$.

Synaptic weights were set randomly in the range from 20 to 30. Spike generation on the presynaptic neuron leads to a sharp increase in synaptic current on the postsynaptic neuron. After the spike it decays exponentially. It induces changes in the concentration of the synaptic neurotransmitter $y_{i,j}$ evolved according to the following equation:

$${\displaystyle \frac{d{y}_{i,j}}{dt}}=-\frac{y_{i,j}}{{\tau }_y}+b_y\theta (t-t_{s{p}_i})$$

(3)

where $t_{s{p}_i}$ determines the times of successive presynaptic spikes, parameter ${\tau }_y$ defines the relaxation time constant, and parameter $b_y$ describes the fraction of neurotransmitter release during spike generation. The parameters in Equation (3) were taken as follows: ${\tau }_y=4 \mathrm{ms},b_y=1$.

2.1. Average Activity Level

Since the activity of the extracellular matrix occurs over much longer timescales (on the order of minutes to hours [28]) compared to the generation of spikes (on the order of milliseconds) in neurons, we need to introduce a variable for the average activity of the neuron, following the approach adopted by the authors in earlier work [29]:

$${\displaystyle \frac{d{Q}_i}{dt}}=-{\alpha }_Q{Q}_i+{\displaystyle \frac{{\beta }_Q}{1+exp(-V_i/k_Q)}}$$

(4)

where ${\alpha }_Q$ is the rate constant, ${\beta }_Q$ is the scaling factor that satisfies the condition $0<{\alpha }_Q<{\beta }_Q$, and $k_Q$ is the slope parameter, $k_Q<1$. The parameters in Equation (4) were taken as follows: ${\alpha }_Q=0.001$ ms, ${\beta }_Q=0.01$ ms, and $k_Q=0.01$.

2.2. Extracellular Matrix Dynamics

To describe the dynamics of the extracellular matrix of the brain (ECM), we adopted the approach proposed by the authors in the following works [29,47,48]. The model of ECM dynamics uses the concept of activity-dependent activation functions, which are commonly employed for phenomenological descriptions of neuronal excitability, such as gating functions for voltage-gated channels in the Hodgkin–Huxley formalism [33]. In order to reduce the dimensionality of the system and increase the computational efficiency for large-scale modeling while preserving the basic modulation of neuronal activity, we used the reduced form of the ECM dynamics model proposed in the papers [47,48]. The key variables describing ECM activity are ECM concentrations, $ECM$, and protease concentrations, P. Thus, the ECM dynamics model can be described by the following system of ordinary differential equations:

$$\left\{\begin{array}{c}{\displaystyle \frac{dEC{M}_i}{dt}}=-({\alpha }_{ECM}+{\gamma }_{P}P)EC{M}_i+{\beta }_{ECM}{H}_{ECM}\left(Q_i\right),\hfill \\ {\displaystyle \frac{d{P}_i}{dt}}=-{\alpha }_P{P}_i+{\beta }_P{H}_P\left(Q_i\right)\hfill \end{array}\right.$$

(5)

where the parameters, ${\alpha }_{ECM,p}$ determine the rate of spontaneous degradation of ECM and proteases concentration, respectively; parameters ${\beta }_{ECM,p}$ describe the rate of formation of ECM and proteases depending on neuronal activity; $H_{ECM,p}$ sigmoid activation functions [33,49] for ECM and proteases concentrations in the following form:

$$H_x=x_0-{\displaystyle \frac{x_o-x_1}{1+exp(-(Q_i-{\theta }_x)/k_x)}}, x=ECM,P$$

(6)

The values of the parameters in Equations (5) and (6) were determined as follows: ${\alpha }_{ECM}=0.001\mathrm{ms}, {\beta }_{ECM}=0.01\mathrm{ms}, {\gamma }_P=0.1\mathrm{ms}, {\alpha }_P=0.01\mathrm{ms}, {\beta }_P=0.01\mathrm{ms},$$EC{M}_0=0,$ $EC{M}_1=1, k_{ECM}=0.15,{\theta }_{ECM}=0.16, {\theta }_P=0.17, k_P=0.05, P_0=0, P_1=1$.

2.3. Extracellular Matrix Modulation of Neural Activity

It is known from experimental data that the extracellular matrix of the brain can influence synaptic transmission through synaptic scaling [50]. The effect of synaptic scaling is associated with a change in EPSC currents. In a previously proposed work [29], we considered the influence of the extracellular matrix of the brain through synaptic scaling on a single neuron by changing the amplitude of postsynaptic currents. In the proposed model, we consider the neural network and the resulting influence of the extracellular matrix of the brain through synaptic scaling of the synaptic weights of glutamatergic synapses. This can be taken into account in the model for glutamatergic synapses as follows:

$$I_{sy{n}_i}=\sum _{i=1}^M{y}_{i,j}{w}_{i,j}(1+{\gamma }_{ECM}EC{M}_i)$$

(7)

where $I_{sy{n}_i}$ is the summation of all synaptic currents of postsynaptic neuron, $w_{i,j}$ is the weight for glutamatergic synapses between neurons, ${\gamma }_{ECM}$ is the coefficient of ECM influence on synaptic connection.

In our case, we consider the effect only on excitatory neurons, while the inhibitory ones serve classical pre to postsynaptic information transmission.

2.4. Neural Network

The spiking neural network consists of 300 neurons, with a ratio of excitatory to inhibitory neurons of 4:1. Neurons are connected all-to-all with a probability of connection for glutamatergic synapses of 5% and for GABAergic synapses of 20%. Since the model uses the mean-field approach to describe changes in the main neuroactive substances (neurotransmitter, ECM, and proteases), we do not separate the effect of ECM on a group of neurons or a group of neurons on ECM, but introduce into the description of each neuron its own dynamics for neurotransmitter and ECM and proteases concentrations.

2.5. Numerical Simulation and Data Analysis Methods

Numerical integration was performed using the Euler method with a step of 0.01 ms. Numerical methods and data analysis were implemented in Python programming language [51] using Pandas library [52] for data processing and analysis, Brian2 [53] for model simulation, Matplotlib, Seaborn [54], and Scipy [55] libraries for data visualization and analysis, and the Detecta library [56] to detect peaks in data based on their amplitude and other features.

The population activity of neurons, denoted as $A\left(t\right)$, is the population firing rate calculated as the sum of neuron spike responses on a time-sliding Gaussian-shaped window. For the Gaussian window, the width parameter specifies the standard deviation of the Gaussian, which in our case is set to 30 ms.

Bursts were detected from the population activity signal, $A\left(t\right)$, using the Detecta library [56] with the following parameters: $mph=15$-minimum peak height and $mpd$ = 10,000-minimum peak distance.

The power spectrum of the population activity of neurons was calculated using the Fourier transform in the Numpy library. The calculation algorithm for this type of data is described in the work [57].

3. Results

Figure 1 illustrates a schematic diagram of the model. The neurons (marked in blue in the layer on the left in Figure 1) in the network are connected “all-to-all” with a probability of connection of 5 percent for glutamatergic synapses and 20 percent for GABAergic synapses. Each synapse formed by the pre and postsynaptic terminals of neurons has its own local dynamics, which is influenced both by connections pertaining to the presynaptic neuron from other neurons, and by molecules of the extracellular matrix of the brain with proteases (the layer on the right in Figure 1a). When an action potential (spike) is generated in the presynaptic neuron, the neurotransmitter is released from the presynaptic terminal, leading to the formation of a postsynaptic current on the membrane of the postsynaptic terminal. The spiking frequency of a neuron integrates its average activity (Figure 1b). With a low average neuronal activity, the release of extracellular matrix molecules from the presynaptic terminal will predominate. As they accumulate and act on postsynaptic currents, a gradual increase in the average neuronal activity occurs, resulting from the effect of synaptic scaling. When the threshold level is reached, proteases begin to be actively produced, which break down the molecules of the extracellular matrix of the brain, stabilizing neuronal activity at the optimal level. In order to comprehend the dynamics of proteases with respect to the average neural activity Q, we have represented the steady-state values of $P_{\infty }$ for various Q levels in Figure 1c. The activation function dictates these values, which can be expressed as $P_{\infty }={\beta }_P{H}_P(Q,P_0,P_1,{\theta }_P,k_P)/{\alpha }_P$. To understand the formation of bursting dynamics in a neural network, we consider three cases: (1) in the absence of ECM modulation of neuronal activity (${\gamma }_{ECM}=0$), (2) in the presence of weak ECM modulation of neuronal activity (${\gamma }_{ECM}=1$), and (3) in the presence of strong ECM modulation of neuronal activity (${\gamma }_{ECM}=5$). The characteristics of neural activity below were obtained by simulating the model for 100 s.

First, we consider the case of solely spiking neuron network dynamics, e.g., for ${\gamma }_{ECM}=0$. Figure 2 illustrates that the spikes of neurons in the raster diagram appear asynchronously in some irregular manner. The network is excited by the uncorrelated noise drive $I_{ext}$ and no synchronous population events happen in this case.

It can be seen that the population activity signal is low in amplitude (up to 10 Hz), which also affects the power spectrum of population activity, $A\left(t\right)$ (Figure 3).

The distribution of interspike intervals (ISI) in this case is as follows (Figure 4). The coefficient of variation for this case is equal to 3.69.

In the second case, with a weak effect of ECM on neuronal activity ${\gamma }_{ECM}=1$, synchronization of neurons begins to occur with the appearance of irregular bursts (Figure 5).

As can be observed, the power spectrum of population activity, $A\left(t\right)$, has increased by an order of magnitude (Figure 6), which is also evident in the amplitudes of population activity, $A\left(t\right)$, in Figure 5. Based on the dynamics of protease molecules concentration, it can be observed that as the amplitude of population activity increases, the concentration of protease molecules also increases and it decreases with a decrease in the amplitudes of population activity, or the formation of large interburst intervals (more than 300 ms).

The synchronization of neurons resulting from the influence of ECM on neuronal activity naturally led to a shift in the distribution of interspike intervals (Figure 7) towards the left, with a noticeable increase in the number of interspike intervals less than 100 ms. In this case, the coefficient of variation is reduced to 2.59.

The increase in the number of long (more than 50 ms) interspike intervals was a consequence of the appearance of bursts and their irregular dynamics.

In this case, the distribution of bursts amplitudes and interburst intervals (IBI) for this case will be as shown in Figure 8 and Figure 9 respectively.

In the last case with a stronger effect of the ECM,${\gamma }_{ECM}=5$, on neuronal activity, the burst dynamics becomes more regular (Figure 10). It can be observed that the amplitude of population activity increased by an order of magnitude, which is associated with a high spike activity of neurons. At the same time, the concentration of proteases reaches its plateau and does not decrease further due to the high population activity.

The network is experiencing a high firing rate, which is reflected in the power spectrum of population activity, $A\left(t\right)$, (Figure 11), which becomes orders of magnitude larger than with a weak impact.

The distribution of interspike intervals (ISI) in this case is as follows (Figure 12). It can be seen that the distribution of ISI has shifted even further to the left, with an increased number of ISIs less than 50 ms. The coefficient of variation has also increased to 8.45. The distribution of bursts amplitudes and interburst intervals (IBI) for this case will look in accordance with the Figure 13 and Figure 14, respectively.

It can be seen that most of the bursts have an amplitude up to 250 Hz (Figure 13). The increase in the number of bursts was also reflected in the increase in the number of interburst intervals (Figure 14). At the same time, most bursts have interburst intervals of 200 ms.

To study the dependence of the impact of ECM on neuronal activity, general characteristics of realizations were chosen: the mean number of spikes (Figure 15) and the mean frequency of bursts (Figure 16). The data in Figure 15 and Figure 16 were obtained from 100 s realizations by averaging 5 experiments at each point.

It can be seen that as the effect of ECM on neuronal activity, ${\gamma }_{ECM}$, increases, the mean number of spikes grows exponentially (Figure 15), which is characteristic of a high-level firing rate in case of synchronization. In this case, the mean frequency of bursts stabilizes at the value of the parameter ${\gamma }_{ECM}$ equal 2.

4. Discussion

We investigated the effect of ECM activity on the synchronization of neuronal activity in the network. The results confirmed our earlier assumption that there may be various dynamic modes of modulation of the power of the synaptic scaling effect [47]. Previously, using a mean-field model of a tetrapartite synapse, we showed [58] that ECM can change the frequency and duration of bursts, which was confirmed by a more detailed network model.

The ECM activity model included the following assumptions: (a) the synthesis of ECM molecules and enzymes that decompose ECM is controlled by the level of neuronal activity, and (b) changes in the ECM level can, in turn, modulate neuronal activity either in an excitatory way.

It should be noted that the proposed model is a radical simplification of the cognitive processing that occurs in the real brain. However, it is expected to be quite predictive since it includes the experimentally observed effect [50]. The study of the effect of ECM on neuronal activity requires complex chronic experiments, which makes the construction of predictive models relevant. This model can provide insight into the role of the ECM in epileptogenesis [35,36,59], and memory impairment associated with ECM remodeling in neurodegenerative diseases [60,61,62].

In the proposed model, we focused on the consideration of one of the main ECM functions observed in the glutamatergic synapse. Further complication of the model may be associated with the introduction of ECM regulation of the action potential generation threshold, as well as modulation into the GABAergic synapse, which will probably allow for obtaining a more extensive repertoire of dynamic modes in network models.

5. Conclusions

A new model of a spiking neuron network interacting with an active extracellular field imitating brain extracellular matrix is proposed. The model is based on the effect of synaptic scaling caused by ECM activity, which leads to a change in synaptic weights. It was found that the weak effect of ECM on neuronal activity leads to population bursting dynamics. Conversely, a strong effect of ECM on neuronal activity leads to a stronger level of population synchrony and the formation of more regular burst dynamics, which in turn is also reflected in a higher firing rate.