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The authors have declared that no competing interests exist.

Experimental studies highlight the important role of the extracellular matrix (ECM) in the regulation of neuronal excitability and synaptic connectivity in the nervous system. In its turn, the neural ECM is formed in an activity-dependent manner. Its maturation closes the so-called critical period of neural development, stabilizing the efficient configurations of neural networks in the brain. ECM is locally remodeled by proteases secreted and activated in an activity-dependent manner into the extracellular space and this process is important for physiological synaptic plasticity. We ask if ECM remodeling may be exaggerated under pathological conditions and enable activity-dependent switches between different regimes of ECM expression. We consider an analytical model based on known mechanisms of interaction between neuronal activity and expression of ECM, ECM receptors and ECM degrading proteases. We demonstrate that either inhibitory or excitatory influence of ECM on neuronal activity may lead to the bistability of ECM expression, so two stable stationary states are observed. Noteworthy, only in the case when ECM has predominant inhibitory influence on neurons, the bistability is dependent on the activity of proteases. Excitatory ECM-neuron feedback influences may also result in spontaneous oscillations of ECM expression, which may coexist with a stable stationary state. Thus, ECM-neuronal interactions support switches between distinct dynamic regimes of ECM expression, possibly representing transitions into disease states associated with remodeling of brain ECM.

Understanding the principles and mechanisms of information processing in the central nervous system is among the main objectives of neuroscience. For a long time, the main role in this process was assigned to neurons. Recent studies have shown that, in addition to neurons, an important role in the processing of information also belongs to glial cells and to the ECM [

The neural ECM, particularly in the form of perineuronal nets, is accumulated during the critical period of postnatal development [^{2+} influx into neurons through interaction between hyaluronic acid and L-type calcium channels (L-VDCC) [

A phenomenological model describing the homeostatic regulation of neuronal activity by ECM molecules was first proposed by Kazantsev and colleagues [

Reductions of the original model of ECM dynamics were done in order to enable the analytical tractability of the resulting model. The polarity of ECM-neuron interactions was changed according to newly available experimental data, showing that that fewer spikes are generated after ECM attenuation due to activation of SK channels [

The processes of ECM synthesis and degradation in a neuronal network are described by the phenomenological approach developed in [_{inf}(_{inf}(

Here the activation functions _{Z,P,R} all assumed to have a sigmoid shape. An increase in the protease concentration _{Z*} = _{Z} + _{P}_{inf} can be approximated by a linear function of _{inf}(

The presented first-order relaxation kinetics model for ECM proteins and proteases concentrations can be viewed as an approximation of a more detailed model of ECM degradation and remodeling (e.g., developed in [

Let us consider the case when the ECM-neuron interaction feedback loop involves either clustering of Kv potassium channels (inhibitory ECM effect) or inhibition of small-conductance calcium-activated SK potassium channels (excitatory ECM effect). In these cases, ECM-neuron interactions are independent of postsynaptic ECM receptor concentration

We assume the effect of ECM concentration on neuronal firing rate might be approximated by a linear dependence _{inf} = _{0} + _{Q}

Let us first qualitatively show that ECM concentration might be bistable in this system regardless of the sign of _{Q}. The equilibrium curves in the ECM-concentration firing rate phase plane (

Both panels show the existence of bistable solutions regardless of the polarity of ECM-neuron interactions. The intersections of the nullclines determine the equilibria of the system: blue points are stable, red points are unstable.

It is apparent that there are cases of bistability, which correspond to the line _{inf} = _{0} + _{Q}_{inf} curve in three points, two stable and one unstable stationary solutions, correspondingly. Note that depending on the sign of the _{Q} parameter, which controls whether ECM influence on neurons is inhibitory or excitatory, the bistability effect is induced by different mechanisms. When the ECM-neuron interaction is excitatory, and hence the slope of the _{inf}(_{inf}(_{inf}(_{P} = 0) would be enough to yield a set of bistable solutions. On the other hand, if the ECM-neuron effect is inhibitory (negative _{Q}), bistable solutions only exist in the presence of the bump in the equilibrium curve _{inf}(_{inf} is smaller at higher firing rates compared to the intermediate range of _{P}).

In biophysical terms, we predict that if the prevalent regulation cascade determining ECM-neuronal interactions restrains neuronal excitability, then ECM bistability can only be implemented if proteases demonstrate a strong effect on ECM degradation. If ECM-neuronal interactions support neuronal excitability, the bistability effect does not depend on the strength of protease-ECM interaction and might be implemented even in the absence of protease-dependent ECM degradation.

Let us more closely consider the case of excitatory ECM-neuron interactions (_{Q} _{Z}.

The number of equilibrium points is determined by the number of intersections of the nullclines _{Z}: _{Z} = 5.68, _{Z} = 6 and _{Z} = 6.4. It is apparent that changes in _{Z} only influence the curve _{Z} = 5.68, the intersection of the curves determines the unique equilibrium of the system that is shown as the green point in _{Z}, the upper part of the _{Z} = 6, three intersection points exist. Particularly, in _{Z} leads to the case with one intersection of the nullclines. In _{Z} ∈ (_{Z}^{L}, _{Z}^{R}) where the left boundary corresponds to coincidence of blue and red points, while the points denoted by red and purple colors coincide at _{Z} = _{Z}^{R}.

The intersections of the nullclines determine the equilibria of the system: green (focus) and blue (node) points are stable, red (saddle) and purple (focus) are unstable. Biologically meaningless areas are indicated as shaded domains. Other parameters are given in

Parameter | Values |
---|---|

_{0}[arb.u.], _{Q}[arb.u.] |
5, 0.23 |

_{Z} [ms^{−1}], _{Z} [arb.u.], _{Z} [ms^{−1}] |
0.0001, 0.15, 0.01 |

_{P} [ms^{−1}],_{P} [ms^{−1}],_{P} [arb.u.],_{P}[arb.u.] |
0.001, 0.001, 6, 0.05 |

To investigate the stability of these equilibrium points we consider the Jacobi matrix:
_{*}, _{*}) obtained from the following system:

Finding the eigenvalues of the matrix (8) as the roots _{1} and _{2} of the characteristic equation:
_{*}, _{*}) for various _{Z} can be determined. Particularly, in _{*} values obtained for various values of the ECM production threshold _{Z}. Different types of equilibrium points are denoted by different symbols. In addition to stable stationary states, there might exist oscillatory regimes as well, with corresponding limit cycles in the phase space of the system. Blue curves in _{min} and maximal _{max} values, which _{Z}. The red curves denote the same, but for unstable limit cycle. The differences _{max}_{min} and _{max}_{min} for stable limit cycles as functions of _{Z} are presented in

Application of an external stimulus (e.g. a spontaneous increase or decrease in neural activity) may induce dynamical switches between activity states. This is an imposed change in neural activity. Parameter value for _{Z} = 3.75. Other parameters are given in

The physical timescale values of the observed ECM oscillations are quite extended in our model since the key assumption is that ECM dynamics is much slower as compared to neuronal dynamics. Experimentally observed changes in ECM concentration may be on the timescale of hours to days [

In the case when the prevalent mechanism of ECM-neuron interactions is through synaptic scaling, the dynamics of ECM receptors might influence ECM dynamics in general. As mentioned above, typically the characteristic timescales of ECM receptor dynamics is significantly shorter than that of ECM molecules and proteases, so that _{inf}(_{inf}(_{inf}). Assuming that the stationary firing rate level scales linearly with the product _{0} − _{R}

Another limit case is when the dynamics of ECM receptors is slow even in comparison to characteristic timescales of ECM remodeling (e.g. the period of ECM concentration oscillations), when the value of

In summary, we have investigated ECM concentration in a mathematical model of ECM-regulated modulation of neural activity. The model is based on the following key assumptions: (a) synthesis of ECM molecules and ECM-degrading enzymes is controlled by the level of neuronal activity, (b) changes in ECM levels may, in turn, modulate neuronal activity, in either excitatory or inhibitory manner, depending on the prevailing mechanism of ECM-neuronal interaction. Mathematically, the model can be reduced to a set of two or three coupled differential equations, depending on the assumptions concerning the nature of ECM-neuronal interactions and characteristic timescales of postsynaptic ECM receptor production. The inhibitory effect of increased ECM levels on neural activity was observed to induce protease-dependent bistable dynamics, while the excitatory effect of ECM-neuronal interaction resulted in a richer repertoire of observable dynamical states. We found that for the excitatory ECM-neuron interactions, e.g. involving the inhibition of SK-channels or synaptic upscaling, the ECM concentration levels may exhibit different activity regimes, ranging from neural firing-induced protease-independent switching between stationary states of the ECM concentration to spontaneous ECM oscillations, which might coexist with a stationary concentration level. In terms of neuronal activity, this means that there are different dynamical modes of ultra-slow firing threshold modulation or modulation of the power of the synaptic scaling effect. Development of more detailed network-based models of neural activity subjected to these ultra-slow modulations might predict the functional effects by which changes in the ECM induced by a seizure or emotional stress might persistently alter the activity of neuronal circuits.

Obviously, the next step for the development of the model is to compare its predictions with experimentally observed dynamics of neuronal activity, activities of ECM proteases and expression of neural ECM components. Ca^{2+} imaging and multielectrode arrays can be used in vitro and in vivo to monitor neuronal activity. Live labeling of ECM of perineuronal nets with Vicia villosa agglutinin is possible in vitro to compare ECM expression at two time-points [