Large-scale brain dynamics can be studied in silico with network models (Deco et al., 2011). Local activity can be represented by neuronal mass models (Deco et al., 2008), which coupled together through synapses, time delays and noise (Deco and Jirsa, 2012, Petkoski and Jirsa, 2019) allow the emergence of whole brain activity that can be linked to empirical neuroimaging data (Sanz-Leon et al., 2015). The observable properties (variables) in the population level of a large-scale ensemble are generally explained by statistical physics formalism of mean-field theory (Wilson and Cowan, 1972, David and Friston, 2003, Moran et al., 2007, Wong and Wang, 2006). The models demonstrated a predictive value for resting state activity (Melozzi et al., 2019, Courtiol et al., 2020) , or for seizure genesis and propagation in epilepsy (Proix et al., 2017). Neural mass models have a low enough number of parameters to be tractable and provide general intuitions regarding mechanisms (Jirsa et al., 2014, Jirsa et al., 2017, Deco et al., 2021, Amunts et al., 2020) . Although it is not practical to include all known biophysical parameters, it may be important to include parameters that can have widespread and general effects on neuronal activity (Amunts et al., 2020) .

The concentration of Na+ , K+, Ca2+, and Cl− ions in the extracellular space is a key parameter to consider. Extracellular ion concentrations change dynamically in vivo as a function of the brain state, for example between arousal and sleep (Ding et bioRxiv preprint doi: https://doi.org/10.1101/2021.10.29.466427; this version posted January 6, 2022. The copyright holder for this preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made available under aCC-BY-NC-ND 4.0 International license. 2 Biophysical neural mass model of ion-exchange al., 2016). Changes in extracellular ion concentrations can switch one brain state to another (Ding et al., 2016). Extracellular potassium concentration ([K+]o) plays a central role. Transient changes in ([K+]o) can have large effects on cell excitability and spontaneous neuronal activity (Amzica et al., 2002, Ransom et al., 2000, Cressman et al., 2009) , a result consistently reported in modeling studies (Bazhenov et al., 2004, Park and Durand, 2006, Fröhlich et al., 2008) . Increases in [K+]o are tightly controlled by astrocytes, which can efficiently pump [K+]o and distribute it via their syncytium to prevent hyperexcitability (Breslin et al., 2018, Nwaobi et al., 2016, Crunelli et al., 2015, Hansson et al., 2000, Bedner and Steinhäuser, 2013) . Saturation or lack of efficiency of this buffering mechanisms is often linked to a pathological state. Detailed single neuron models demonstrate that continuous increases in [K+]o can lead to different firing states, from tonic, to bursting, to seizure-like events and depolarization block (Cressman et al., 2009, Depannemaecker et al., 2021) . In such detailed models, the buffering action of astrocytes is represented by a parameter named [K+]bath (Cressman et al., 2009, Breslin et al., 2018, Ullah et al., 2009, Nwaobi et al., 2016, Depannemaecker et al., 2021) . Our goal is to use a similar approach but at the neural mass model level, to be incorporated to study whole brain dynamics.

Phenomenological neuron models have been studied to understand the macroscopic dynamics of neuronal populations (Izhikevich and Edelman, 2008) or to provide statistical descriptions of neuronal networks (Wilson and Cowan, 1972, Deco et al., 2008, Montbrió et al., 2015, Montbrió and Pazó, 2020). Statistical population measures, such as the firing rate, can be used to assess macroscopic dynamics (Shriki et al., 2003, Deco et al., 2011, Roxin et al., 2005, Luke et al., 2013, Jirsa et al., 2014, Proix et al., 2014, Sanz-Leon et al., 2015, Wendling et al., 2016, Ashwin et al., 2016) . Recent studies combine local neuronal dynamics with statistical descriptions in terms of the mean membrane potential and the firing rate in quadratic integrate-and-fire (QIF) neurons (Montbrió et al., 2015, Devalle et al., 2018, Montbrió and Pazó, 2020). Neural mass models and large scale brain dynamics models are also used for analyzing and identification of chaos in brain signal (Baladron et al., 2012, Bandyopadhyay and Kar, 2018) , epileptic seizure transmission (Spiegler et al., 2011, Touboul et al., 2011, Nevado-Holgado et al., 2012, Hocepied et al., 2013, El Houssaini et al., 2015a) , phase oscillation (Laing, 2017, Byrne et al., 2017, Petkoski and Stefanovska, 2012) , evolution of network synchrony (Bandyopadhyay and Kar, 2018, Petkoski and Stefanovska, 2012) and several other problems. Here we approximate ion-exchange dynamics by a step-wise QIF model with two slow timescale biophysical variables. We demonstrate that the distribution of the neurons’ membrane potentials can be described by a Lorentzian ansatz (LA). The continuity equation is solved to give rise to a mean-field model with the same probability distribution of membrane potentials as the LA. The mean-field model is exact for non-autonomous ion concentration variables and provides a mean-field approximation within the thermodynamic limit, i.e., for a locally homogeneous mesoscopic network. We thus demonstrate a mean-field approximation for an all-to-all coupled network of heterogeneous neurons to approximately capture the behavior of ion-exchange-driven neuronal dynamics. Considering different distributions of heterogeneous input current, we obtain a mean-field approximation, described as a set of ordinary differential equations that fully described the macroscopic states of the recurrently connected spiking neurons, in various dynamical regimes that can be linked to different healthy and pathological states.

In this paper we have derived a mean-field approximation of locally homogeneous network of Hodgkin–Huxley type neurons (HH neuron) with heterogeneous quenched external input η (Eq. (28), and Eq. (29)). The appropriateness of mean-field approximation is validated by comparing the simulation results of the large network of coupled HH neurons with the mean-field model Eq. (29) which appears to be quite robust under stochasticity and different type of distribution of heterogeneous quenched external input η (Fig. 4). Our model demonstrates different types of spiking and bursting behavior as well as resting-state (Fig. 6) and multistability in the distinct regime of external potassium bath ([K+]bath). These kinds of neuronal activities are quite observable in large-scale brain dynamics which by the means of this novel mean-field model get connected with the ion-exchange mechanism and the ion concentration states in the cellular space. The bifurcation analysis of the mean-field model is carried out to identify relevant parameter regime which is responsible for different types of spiking and bursting behavior, resting-state, and multistability features in the mean-field model (Fig. 5). Moreover, it is observed that even if in the resting state (healthy) network certain kinds of external stimulus current could generate transient neuronal bursts (in terms of transition between upstate and downstate) which are validated by parallel simulation of the large network of coupled HH neurons and the mean-field model (Fig. 7). This result could be quite significant in terms of explaining the neuronal activities during brain stimulation. Furthermore, the behavior of two of such mean-field representations under the aforementioned conductance-based coupling (Eq. (9)) are demonstrated (Fig. 8) and validated with the simulation of two groups of neuron with different parameter. It could serve as a baseline to explain how epileptic seizures could propagate through some healthy brain regions as a result of their coupling with epileptogenic regions.

The simulation of Hodgkin–Huxley type single neuron dynamics Eq. (1) driven by an ion-exchange mechanism, has revealed that the parameter [K+]bath is the most important parameter for describing the dynamics. Previous studies showed that changing the value of the concentration of the external potassium bath is responsible for qualitative changes in the dynamical behavior of a single neuronal system (Depannemaecker et al., 2021), particularly discovering that in the parameter regime [K+]bath = [6.01, 6.875] multistability occurs.

Reproducing the bifurcation analysis by a numerical continuation revealed two stable fixed points and a stable focus in this regime for [K+]bath. We also observed a Hopf bifurcation at [K+]bath = 7.68. At higher values of [K+]bath he stable fixed-point behavior of the dynamics changes into limit cycle behavior and the dynamics shows the bursting characteristics of neurons. The single neuron model was also shown to be able to produce different dynamic behaviors of single neuronal activity, such as epileptic bursts, status epilepticus, resting state, etc. In this study we developed a mean-field dynamics system based on a biophysical single neuron model Eq. (1) to describe the network behavior of biophysical neurons within the thermodynamic limit.

To be able to apply convenient mathematical (analytical) methods in this study, the velocity of the membrane potential of a single neuron described by Eq. (1) was approximated as a stepwise quadratic function because geometrically it resembles a combination of two inverted parabolas. The validity of this approximation is shown in Fig. 2 where the time derivative of V in Eq. (1) is compared with that of a stepwise quadratic approximation. The appropriateness of the approximation of the membrane potential dynamics is represented for different values of the other state variables and parameters, such as n, ∆[K+]i, and [K+]bath in Fig. 2(a), 2(b), and 2(c), respectively.

bioRxiv preprint doi: https://doi.org/10.1101/2021.10.29.466427; this version posted January 6, 2022. The copyright holder for this preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made available under aCC-BY-NC-ND 4.0 International license. 12 Biophysical neural mass model of ion-exchange

The simulation of the mean-fields dynamics reveals that [K+]bath serves as a bifurcation parameter of the dynamics. Scanning through its values in the biophysical range changes the behavior of the dynamics from a stable fixed point to an oscillatory limit cycle type of behavior and back to a stable fixed point. To carry out the bifurcation analysis, numerical continuation was applied, and we found that a Hopf bifurcation exists at [K+]bath = 7.68 and a reverse Hopf bifurcation could be found at [K+]bath = 25.02. In between these two values of [K+]bath, the dynamics manifests an oscillatory limit cycle behavior, whereas outside of this range the dynamics of the system shows a stable fixed-point behavior.

Fig. 5 presents the 3-dimensional bifurcation diagram (Fig. 5(a)) in v and ∆[K+]i with respect to [K+]bath, and the 2-dimensional projections (Fig. 5(b)) in the v dimension for the mean-fields model Eq. (29), indicating the Hopf bifurcation. Multistable behavior of the mean-field was found in this model for [K+]bath > 5.971, as it is demonstrated in Fig. 5. Plots in Fig. 5(c), and Fig. 5(e) illustrates the time series of the dynamics for healthy ([K+]bath = 7.65), and bursting ([K+]bath = 11.5, 18.5) states respectively. whereas Fig. 5(d), and Fig. 5(f) represent the phase space trajectories for corresponding healthy, and bursting states. Together, they indicate the up state, down state, and epileptic state for different values of [K+]bath. When [K+]bath is smaller than the Hopf bifurcation, the downstate is found to be a stable fixed-point, which destabilizes into spiking, and bursting behaviors and gradually into epileptic bursts at the bifurcation. On the other hand, the upstate emerges at [K+]bath = 5.971, which is a stable fixed point of the vfie-dimensional system (Eq. (28), and Eq. (29)). However, if the fast bioRxiv preprint doi: https://doi.org/10.1101/2021.10.29.466427; this version posted January 6, 2022. The copyright holder for this preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made available under aCC-BY-NC-ND 4.0 International license.

Biophysical neural mass model of ion-exchange 15 subsystem is decoupled and the slow variables are considered to be parameters, the fast subsystem in the upstate undergoes bifurcation, demonstrating a limit cycle, a stable spiral, and a stable fixed point as the value of the slow variable decreases.

The model can also demonstrate different neural activities, as demonstrated in Fig. 6. For different regimes of [K+]bath the mean-field model is capable of producing a large set of brain activities, such as a spike train (Fig. 6(a)), tonic spiking (Fig. 6(b)), bursting (Fig. 6(c)), a seizure-like event (Fig. 6(d)), a status epilepticus -like event (Fig. 6(e)), and a polarization block (Fig. 6(f)).

The general quest of modern neuroscience is understanding and explaining the mechanisms of different brain activities like perception, memory and decision making; but also more generally how the brain functions for healthy individuals during rest, and how it deviates from its healthy state in case of different diseases and disorders. Interestingly most of these brain activities are accessed through measurements of brain dynamics that manifest the collective electrical activities of a huge population of neurons demonstrating some inter-related electrical behavior in different brain regions. Therefore, the brain activities in general can not be explained in microscopic level that is as the mechanism of a single neuron rather they emerge in macroscopic level as a result of the interaction between large population of neurons. Also, besides the recent progress (Markram et al., 2015),it is still not possible to compute the behavior of a few billions of such neurons (which comprises a mammalian brain), and even if it was, it remains questionable what knowledge would we gain from such a complex system (Frégnac, 2017). Hence, the most realistic way to model the brain activities is to group a number of neurons together and write the mathematical descriptions for a group of neurons or large population (Sanz-Leon et al., 2015), using formalism from statistical physics and mathematics. In this case we measure some property of the whole population, such as the mean membrane potential, rather than the activity of an individual neuron. Some of these properties do not even exist for an individual component, for example temperature or bioRxiv preprint doi: https://doi.org/10.1101/2021.10.29.466427; this version posted January 6, 2022. The copyright holder for this preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made available under aCC-BY-NC-ND 4.0 International license.

Biophysical neural mass model of ion-exchange 17 pressure in case of molecular ensembles, order parameter (Kuramoto, 1984) and mean ensemble frequency (Petkoski et al., 2013) for coupled oscillators, or more specifically the firing rate in case of neuronal population (Gerstner and Kistler, 2002). However, to the best of our knowledge till the date there is no theory that can explain the behavior of large population of neurons (brain areas) from the perspective of the driving mechanism of the neuronal activities, that is the ion-exchange and transportation dynamics in the cellular level. Different pathological trials and experiments have already revealed that changes in ion concentration in brain regions could result in different brain dysfunction but the pathway of these phenomena is still unclear.

In this study we developed a biophysically inspired mean-field model for a network of all-to-all connected, locally homogeneous Hodgkin–Huxley type neurons which are characterized by ion-exchange mechanism across the cellular space. The intermediate approximation into the step wise quadratic function allowed to apply analytical formalism to the complex Hodgkin–Huxley type equations and derive the distribution of membrane potential as Lorentzian distribution. The Lorentzian ansatz makes the mean-field approximation to be analytically tractable, and from the simulation of network behavior of such neurons it is evident that the mean-field model captures the dynamic behavior of the network, while being quite robust for different distributions of heterogeneity and even for noise of small magnitude. This model relates the mechanism of the biophysical activity of ion-exchange and ion channel transportation to the phenomenology of the whole brain dynamics. The major characteristic feature of this model is the analytical bridging between intracellular and extracellular potassium concentrations which in principal act as an adaptation (a biophysical regulation that changes the electrical activity of neurons, acting on a relatively slow time scale), and fast electrical activities in different brain region demonstrated via brain imaging of different modalities. Specifically, the derived model relates the slow scale biophysical mechanism of ion-exchange and transportation in the brain to the fast scale electrical activities of large neuronal ensembles in terms of mean-field formalism of the membrane potential.

Analysing the model, we found the extra-cellular potassium concentration (potassium bath) and heterogeneity of individual neurons in the population to be two significant determinants of the dynamics. Together they take care of the biophysical state of neuronal population (brain region) in terms of ion concentration and structural diversity. In principle these two factors are major drivers of the neuronal dynamics and brain activities and successfully emerged from our simulations. Moreover, the developed model demonstrates the coexistence of resting state brain dynamics and epileptic seizures depends on different states of biophysical quantities and parameters. This aforementioned analytical formalism demonstrates a class of different neural activities, such as the existence of up and down states during the healthy parametric regimes, which is the hallmark of many mean-field representations of linear (Di Volo et al., 2019, Zerlaut et al., 2018) or quadratic integrate and fire neurons (Coombes and Byrne, 2019, Montbrió et al., 2015) , and rate models (Wong and Wang, 2006). Increasing the excitability in terms of ion concentration, on the other hand leads to appearance of spike trains, tonic spiking, bursting, seizure-like events, status epilepticus -like events, and depolarization block, similarly as in the epileptor (Jirsa et al., 2014, El Houssaini et al., 2015b, El Houssaini et al., 2020) . The mean-field approximation model links the high and low firing rate states (so called up and down state) as well as spiking and bursting behaviors of electrical excitability in neuronal dynamics with the biophysical state of neuronal ensemble (brain regions) in terms of the ion concentrations across the cellular space. The effect of different kind of stimulus current is also analyzed and it is observed that even within the healthy regime, several stimulation, which could either be an external stimuli or some input from some other brain regions, could generate transient spiking and bursting activity in different brain regions. This result is particularly interesting in case of brain stimulation. For example several kind brain stimulation in epileptic patients have already been found to generate epileptic seizures in pathological practice (Fisher and Velasco, 2014, Kahane and Depaulis, 2010) . Our results demonstrate that these phenomena could be analytically reproduced and tracked. Therefore, we assume the derived model could potentially be applied to improve predictive capacities in several type of brain disorders, and particularly in epilepsy.

Using conductance-based coupling between two neuronal masses we also demonstrated that a bursting population of neurons can propagate bursting and spiking behavior to an otherwise healthy population. This result is validated by the symmetry in qualitative behavior found in the simulation of the mean-field model and the corresponding network of Hodgkin–Huxley type neurons. This could lead the path to understand how brain signals propagates as a coordinated phenomena depending on the distribution of biophysical quantities and structural as well as architectural heterogeneity on the complex network structure of the connectome. It would also be very interesting to study how this signal propagation varies among healthy subjects and compared with patients with neurological disorders. For an example this model could serve as a computational base line to understand the core question of epilepsy research that is how epileptic seizures propagate from epileptogenic zone to propagation zone and to precisely identify these brain regions in terms of their biophysical states characterized by the distribution of the relevant biophysical and pathological markers as well as structural properties.

Until now, mean-field models used in large scale network simulations like the Virtual Brain did not take into consideration the extracellular space. This model is a first step towards the integration of biophysical processes that may play a key role bioRxiv preprint doi: https://doi.org/10.1101/2021.10.29.466427; this version posted January 6, 2022. The copyright holder for this preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made available under aCC-BY-NC-ND 4.0 International license. 18 Biophysical neural mass model of ion-exchange in controlling network behavior. We have started with K+, given its known role on neuronal activity. Changes in K+ occur when the brain alternates between arousal and sleep (Ding et al., 2016). Our model, once integrated in a whole brain model, will allow studying such physiological mechanisms. Although a causal relationship is not clearly established, seizures and spreading depression, which is assumed to underlie certain forms of migraine (Tottene et al., 2009, Vinogradova, 2018), are associated with large (>6 mM) and very large (>12 mM) concentrations of K+ (Tottene et al., 2009, Hertz and Chen, 2016). The extracellular concentration of K+ is tightly controlled by astrocytes (Breslin et al., 2018, Nwaobi et al., 2016, Hansson et al., 2000) . Most large-scale simulations do not integrate astrocytes, which make half of brain cells (Hansson et al., 2000, Breslin et al., 2018) . Their functions are altered in most, if not all, brain disorders, in particular epilepsy (Bedner and Steinhäuser, 2013, Crunelli et al., 2015) . Our approach allows formalizing the astrocytic control of extracellular K+. Other ion species also vary during arousal/sleep and seizures, in particular Ca2+ (Ding et al., 2016, Pocock and Kettenmann, 2007, Auld and Robitaille, 2003, Fernandez-Chacon et al., 2001) . A decrease in Ca2+ will decrease neurotransmission and thus change cell to cell communication (Pocock and Kettenmann, 2007, Auld and Robitaille, 2003, Fernandez-Chacon et al., 2001) . Future studies are needed to integrate Ca2+ in neural mass models (NMM).

Our mean-field derivation aggregates a large class of brain activities and behavior patterns into a single neural mass model, with direct correspondence to biologically relevant parameters. This paves the road for brain network models with bottom-up approach incorporating the regional heterogeneity that stems from the structural data features. We believe that mean-field formalism addressing the biophysical information across neurons will tell us more interesting things about the relation of different dynamics of the brain with its biophysical parameters. This would eventually lead to identifying pathologically measurable bio-markers for large scale brain activities and consequently offering therapeutic targets for different brain dysfunctions. bioRxiv preprint doi: https://doi.org/10.1101/2021.10.29.466427; this version posted January 6, 2022. The copyright holder for this preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made available under aCC-BY-NC-ND 4.0 International license.

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