Equations of the JR model

The JR model represents a local cortical circuit of three interconnected populations captured by six differential equations, two for pyramidal projection neurons Eqs (1A–1B), two for excitatory Eqs (1C–1D) and two for inhibitory interneurons Eqs (1E–1F), which form feedback loops. Neural populations are characterized by a second-order differential operator that converts the average incoming spike rate into the average membrane potential, whereas a non-linear, sigmoidal, function is applied to transform the average membrane potential into the average output spike rate. The state variables y₀, y₁ and y₂ describe the outputs of pyramidal, excitatory, and inhibitory postsynaptic potential blocks (EPSP and IPSP) respectively. The postsynaptic potential (PSP) of the i-th pyramidal population is given by the difference PSP_i = y_1,i−y_2,i. We implemented one additional state variable to the tuning model, namely wFIC_i, which describes the adaptive strength of the inhibitory input onto the pyramidal cells of region i in Eq (1B). The graphical representation of the model can be found in the Fig 1.

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Fig 1. The graphical representation of the JR model with the introduced heterogenous inhibitory scaling variable wFIC_i.

The three populations Excitatory Interneurons, Pyramidal Cells and Inhibitory Interneurons are marked in orange, red and blue respectively. The sigmoidal is denoted with S and h_e(t) and h_i(t) represent the impulse response resulting from the model equations Eqs (1A–1F).

https://doi.org/10.1371/journal.pcbi.1012595.g001

In the following equations, C denotes the SC matrix and C_ij are the entries of that matrix, i.e., the connection weights between regions i and j. The symbol N_ROIs = 84 denotes the number of nodes in the connectome. Analogously, θ_ij is the time delay and is computed from the tract lengths between regions i and j, as well as the conduction speed value, which was set at 5 m/s. Additionally, we introduce I_ext,i—the total external current received by region i, consisting of a constant component μ and a term representing long-range excitatory coupling, which sums over the outgoing firing rates of all regions j, weighted by the corresponding SC element C_ij Eq (3). The parameter values used for simulations are listed in Table 1 and based on physiological considerations in the original Jansen and Rit publication [52]. The amplitudes of the EPSP and IPSP are determined by A = 3.25 mV and B = 22 mV for the excitatory and inhibitory cells, respectively. The corresponding durations of EPSP and IPSP are inversely proportional to a = 0.1 ms^–1 and b = 0.05 ms^–1. As in the original work [52], the firing threshold is given by v₀ = 6 mV. The remaining parameters of the sigmoidal transfer function in Eq (2) were set at v_max = 0.0025 s⁻¹, e₀ = 2.5 s^–1 and r = 0.56 mV^–1. The constant J = 135 corresponds to the number of the synaptic contacts, which is scaled for individual connections with c₁, c₂, c₃ and c₄.

(1A–1F)

(2)

(3)

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Table 1. Parameters and their values.

These values are fixed throughout this work, if not stated differently. EPSP: excitatory post-synaptic potential, IPSP: inhibitory post-synaptic potential, PC: pyramidal cells, EI: excitatory interneurons, II: inhibitory interneurons.

https://doi.org/10.1371/journal.pcbi.1012595.t001

Regimes of the JR model

The JR model exhibits a complex repertoire of regimes and bifurcation points that is dependent on its parameters depicted in Fig 2. The presence of bistability, where two stable states coexist within a specific parameter range, is one of the fundamental features of this model. In the case of the uncoupled JR model one of the regimes generates the ⍺-band (8−12 Hz) oscillatory activity–here referred to as fast LCs with frequencies in the range 10−12 Hz. Another regime gives rise to slower, δ-band (1−4 Hz) and θ-band (4−5 Hz) rhythm oscillations—referred to as slow LCs. In the following we will omit the region index i for the case G = 0, corresponding to an isolated JR model. Given a deterministic setting, as the input μ is increased from I_ext = 0 to I_ext<0.12, the postsynaptic potential increases steadily, and the model’s behavior is determined fixed points (non-oscillatory regime). This state configuration of the system can be described as sub-bistable and corresponds to 0.007<y₀<0.019. Only when input to an individual node reaches I_ext>0.12, the system starts exhibiting bistability between different regimes. The first bistability exists between the low- and high-activity fixed points (FPs) for I_ext∈[−0.012, 0.09], the second one between the low activity FPs and the fast LC for I_ext∈[0.09, 0.113], and the third one between the fast and slow LCs for I_ext∈[0.113, 0.137]. Further increases to the input, I_ext>0.137, place the system in the fast LC and I_ext>0.31 leads to a high-activity FP (non-oscillatory regime), which we jointly refer to as super-bistable regime, where y₀>0.1. Importantly, when coupling multiple JR nodes, the network effects can give rise to dynamics not captured by the bifurcation diagram of a single, uncoupled node. Nevertheless, the bifurcation diagram for the uncoupled node can approximate network effects under the assumption that each node receives a similar slowly varying “mean-field” input with small noise fluctuations, Therefore, the bifurcation diagram of a single node only approximates the behavior of a network node and may not be precise.

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Fig 2. Bifurcation diagrams of a single uncoupled JR model.

A: State variable y₀ of fixed points (FPs) and limit cycles (LCs) vs. constant external input I_ext = μ = const. At low inputs I_ext≈−0.1, one can find a branch of stable FPs (black line) which undergoes a series of bifurcations. The branch folds at I_ext≈−0.1, where a saddle-node bifurcation (SN, black dot) destabilizes it (dashed black line). A second SN follows, without changing the stability. The two SN result in an S-shaped bifurcation structure. The upper branch stabilizes via a subcritical Hopf bifurcation near I_ext≈0, which gives rise to unstable LCs (dashed orange line). The LC branch becomes stable (solid orange line) through a SN of LCs (purple dot) and terminates through a saddle-node on invariant circle bifurcation (SNIC, green cross). The upper stable FP branch destabilizes and stabilizes again through two supercritical Hopf bifurcations, between which one can find another family of LCs. B: Same bifurcation structure showing the postsynaptic potential PSP vs. I_ext. The bifurcation diagram was computed numerically using AUTO-07p [60]. The parameter values are given in Table 1.

https://doi.org/10.1371/journal.pcbi.1012595.g002

Dynamic feedback inhibitory control

Importantly, due to recurrent activation in the network case, the overall sum of incoming activity to a network node can increase significantly compared to the single-node scenario. The increase in the input to multiple single nodes results in an increase in the overall excitation in the system and an effectively reduced repertoire of available regimes that the individual nodes can achieve. Such a system exhibits behaviors associated with high-activity states, making it impossible to reach the regimes requiring weaker inputs. In the case of JR, such configuration places most nodes in the fast LC or, in extreme cases, near the high FP and leads to increased synchronization of the activity across nodes. Another consequence is the decreased variability of derived slow timescale signals such as BOLD, since they are computed from PSPs of low variability. dFIC was conceived as a solution that allows counteracting over-excitatory network effects by locally adjusting the inhibitory strength of nodes to reflect the overall strength of incoming connections from other nodes such that the long-term average output PSP activity of pyramidal cells of each node is fixed at a chosen level.

Our solution splits the simulation process into two parts: the deterministic tuning process and the stochastic post-FIC simulations. In the first step, we implement the tuning process, which dynamically adjusts the weights of the wFIC_i state variables to bring the model’s activity to the desired level. To achieve this, we add two additional, activity-detection equations Eqs (4A–4B) and one equation describing the evolution of the wFIC_i variables Eq (4C), based on a gradient-descent learning rule proposed by Vogels [61], first implemented in large-scale whole-brain networks by Schirner et al. [17]. The activity-detection equations are used to extract a slowly varying average of the two state variables necessary for computing the learning process during the tuning step: and , for averaged PSP output of the pyramidal and inhibitory populations, respectively. Thus, fast oscillations and short-term variability of y_0,i and y_2,i are averaged out. We chose to include such equations to facilitate our analytics by making a fast- and slow-time-scale separation more explicit, as well as in order to disassociate the slow averaging dynamics from the particular form of the learning dynamics, and given that some slow averaging is in all cases necessary to filter out fast noisy fluctuations (in the case of FP targets) or oscillations (in the case of LC). The Eq (4A) provides a time-averaged slow detection of the ongoing firing rate as fitting target, which ensures that wFIC_i variable does not fluctuate or oscillate.

(4A–4C)

The crucial parameter defining the target of the dFIC procedure is , which is the desired value to which the long-term (i.e., at a slow time scale) average activity of the system is being brought to by the tuning process. This value must correspond to one of the attractors (FP or LC) of the uncoupled, original JR model (Fig 2). For point attractors, the value corresponds to their exact location, whereas for LCs the value corresponds to their centers (in practice approximated as the average activity value over several cycles). The η parameter describes the learning rate at which the wFIC_i variable is being adjusted during tuning to reach . The time constant of the detection equations is denoted by τ_d. The parameters η and τ_d can be adjusted depending on the desired target dynamical regime of the model and simulation length. For example, η determines how quickly the gradient in the dFIC algorithm descends. If η is chosen too large, the values of the state variables wFIC_i may oscillate. Once the values of wFIC_i have converged, the tuning is completed and post-FIC simulations can be performed, for which wFIC_i is replaced by a constant parameter pFIC_i Eq (5A).

, where t_k = kdt is the k-th timepoint of the numerical simulation.

(5A–5B)

To obtain pFIC_i, we time-averaged the last L = 3000 ms of the total tuning simulation of T_tune milliseconds of the wFIC_i state variable from the tuning step. The heterogeneous pFIC_i scales the strength of the inhibitory connection for post-FIC simulations. The graphical representation of the tuning process can be found in S2 Fig.

Simulations

To test the range of values and limits of the parameters of the tuning approach, we ran a series of simulations using a small-network connectome, consisting of 4 nodes with random weights and distances. We used the deterministic Heun’s integration scheme for all the combinations of the detection time constant τ_d and the learning rate η with the tuning simulation length T_tune = 250 seconds. This analysis constitutes an intermediate step between the analytical single-node analysis and whole-brain simulations, designed to help determine the values of the two tuning parameters τ_d and η, and to estimate the necessary length of upcoming whole-brain simulations. In every tuning setup, a 15 s transient has been discarded before the tuning started. We have selected two values, namely, and , to tune the system to, each chosen at one of the attractors of the single-node, original Jansen-Rit model, to maintain the system’s dynamics in the proximity of that attractor. For both values, we have adjusted the external input value μ and all of the respective initial conditions, based on the JR model bifurcation diagram of the single node (Fig 2), to allow for faster and smoother tuning toward the desired regime: first, at the low stable FP, close to the bifurcation point towards the slow stable LC at , in which case we set the μ = 0.09 and the randomized initial conditions below the target value, such that the tuning procedure increases the activity of the system to the desired level; second, at the fast, stable LC, close to the bifurcation point towards the slow stable LC at , in which case we set the value of μ = 0.14 and the randomized initial conditions above the target value, so that the tuning procedure decreases the activity of the system to the desired level. We ran small-network simulations for all combinations of the following settings:

• 
• τ_d∈{10, 100, 550, 1000} ms
• η∈{0.00035, 0.001, 0.005, 0.01}
• G∈{0, 1, 10, 25, 100}

Next, to demonstrate the scalability of our solution, we proceed to whole-brain network simulations. We have used the publicly available HCP Young Adult data set, which includes 3T MR imaging data from healthy adult participants (age range 22–35 years) [62]. The average connectome was derived from multimodal neuroimaging data (diffusion-weighted MRI) and T1-weighted MRI using an automatic preprocessing pipeline [59] processed with Desikan-Killiany parcellation [63], composed of 68 cortical gyrally-based regions, and 16 subcortical regions, added according to the subcortical segmentation present in the Freesurfer software [64].

To test the FC fit for different activity targets and corresponding optimal global coupling values, we ran the full two-parameter search for seven low-activity , five high-activity and 30 values of G∈[0, 30], with step of 1, resulting in a total of (5+7)×30 = 360 parameter combinations. All tuning simulations were set up with a deterministic Heun’s integration scheme (T_tune = 4 min, timestep dt = 1 ms) and post-FIC simulations were run using Heun’s stochastic integration scheme (T_p = 3 min, timestep dt = 1 ms) with additive noise σ = 10⁻⁷ Eq (5B). For each of the post-FIC simulations, we obtained the PSPs of pyramidal cells, for which we performed power spectrum analysis, Poincaré mapping analysis for each node and computed the resting state Blood Oxygenation Level Dependent (BOLD) time-series (T_bold = 30 min, repetition time TR = 720 ms) using hemodynamic response functions with the Balloon Model [65]. To compare post-FIC simulations with the original JR setup, we ran respective simulations for all the G values to obtain the simulation results without the dFIC (later referred to as no-FIC simulations). Values of pFIC_i were set to 1 in the no-FIC simulations, while the remaining parameters, including noise_i and μ, corresponded to their post-FIC counterparts.

The introduction of stochasticity has two main consequences. Due to the nonlinearity of the model, it increases the overall input per node, which, especially in case of high G values, causes the constrained activity of the post-FIC system to deviate from the activity level. Additionally, as demonstrated by further results, noise introduces variability to allow the individual nodes to switch dynamical regimes in the proximity of the chosen activity level. Measuring and quantifying switching behavior in neural systems can be done with the help of Poincaré mapping analysis [66,67].

We implemented a method for a Poincaré mapping involving the analysis of a PSP time-series [68–70]. Poincaré maps are derived from the Poincaré sections, representing a projection of a dynamical system’s behavior onto a lower-dimensional space. Here we take the simulated PSP timeseries PSP_i(t) of region i and determine the k-th local maximum PSP_i,k numerically by comparing each value at a given time point with neighboring points in a 100 ms time window. The recurrence map given by PSP_i,k→PSP_i,k+1 yields a point cloud in which each point corresponds to state changes of individual nodes. The method essentially maps the continuous dynamical system into a discrete one, acting like snapshot of changes in the maxima of the signal. Nodes which stay nearby an attractor, be it FP or LC, exhibit fluctuations around that attractor due to noise. Consequently, the points (PSP_i,k, PSP_i,k+1) spread around the attractor–for FP around the PSP value of the FP and for LCs around all PSP value of local maxima within one period (S1 Fig).

Fitting methods

For BOLD, we computed FC matrices using the Pearson correlation coefficient. To assess the fit between empirical and simulated data we have compared our simulation results to both the average FC matrix and to an average functional connectivity dynamics (FCD) distribution derived from the Human Connectome Project Young Adult data set [62]. For each of the parameter combinations, we computed the value of FC of the BOLD signals and calculated the Pearson Correlation between simulated and empirical data (R_FC). Further we utilized average BOLD FC (the mean of the upper triangle of the empirical FC matrix), as a threshold for the fitting procedure of the simulated FCs to avoid over-synchronization. For our BOLD FC fitting, we have discarded the simulations with FC_mean > 0.25, slightly above the average of the empirical FC matrix (FC_mean = 0.23). This process helped us to reject biologically unplausible and over-synchronized FC data (S4 Fig). FCDs were computed using multiple FCs per simulation obtained by sliding window of size of 60 seconds and increment with 1 TR = 720 ms. The fit was quantified by the Kolmogorov-Smirnov distance (1−KSD_FCD) between the distributions of empirical FCD values and each of the simulated FCDs. Additionally, we combined the fits of R_FC, 1−KSD_FCD, and a synchronization threshold into a single multimodal factor (MMF) which is described by Eq (6). The coefficients d₁, d₂ were set at 1, and 0.75, respectively. The selection of coefficients was motivated by the general idea that both R_FC, and 1−KSD_FCD should contribute to the final measure equally despite having different maximum values, with 0.6 and 0.8 respectively. The coefficient d₂ was set at 0.75 to fulfil this requirement. We found this combined method useful for assessing the biological plausibility of our simulated data, but it is worth noting that the threshold value and coefficients are likely dataset specific and should not be used without numerical justification and testing.

(6)

To confirm that no additional simulations are needed for a robust estimation of the MMF measure, we performed a bootstrapping statistical analysis, for each parameter combination (G, ) and for each condition (post-FIC and no-FIC simulations). To that end, we generated a bootstrapping sample distribution by resampling with replacement from an original set of–per parameter combination–MMF values, computed for overlapping windows of a shorter time length (19.2 min that we extracted from the original 30 min simulations. The results show that the MMF values we computed were robust already across such shorter time windows, for more details see S1 Appendix.

Control of a single Jansen-Rit population

In this section we present results on the control of a single JR node in presence of the homeodynamic inhibitory plasticity mechanism dFIC. We outline how dFIC achieves a desired activity level when the external input I_ext = const., since this is a prerequisite for the functioning of dFIC in a network setup, where I_ext is time dependent. The following semi-analytical methodology is based on numerical bifurcation analysis and slow-fast dissection [71,72]. To start with, we note that the full system for an isolated node, described in Eqs (1),(2) and Eq (4), can possess different timescales, depending on the parameter values. The JR dynamics are governed by the time constants 1/a = 10 ms and 1/b = 20 ms. Typical values for the detection time constant τ_d and the learning rate η used in this work are τ_d = 1000 ms and 1/η = 400 (mV)² ms. With this configuration the full system has three timescales. For simplicity however, we assume a two-timescale system, by posing that τ_d and 1/η are large enough, such that (i) and converge to long-term temporal averages and (ii) changes in wFIC are slow, giving time to the JR state variables to converge to the proximity of attractors present at constant wFIC. This assumption allows us to apply slow-fast dissection, to understand the dynamics of the full system (i.e., single node with dFIC), which emerges from the dynamics of the no-FIC system through a slow variation of the wFIC.

In the limit case 1/τ_d→0, η→0, corresponding to no-FIC, wFIC becomes a parameter in the remaining equations Eqs (1) and (2) and become the long-term temporal averages of y₀ and y₂, respectively. To understand the dynamics in this limit case, we average the LCs of the original JR model (Fig 2) over one period, as shown in Fig 3A for wFIC = 1.

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Fig 3. Bifurcation structure of the uncoupled JR model with averaged limit cycles (LCs).

A: Average versus external current I_ext for wFIC = 1. The solid (dashed) black lines correspond to families of stable (unstable) fixed points (FPs). For families of stable (unstable) LCs, the average value of y₀ over one cycle is computed and depicted as solid (dashed) orange lines. B: Bifurcation structure in space. The bifurcation structure of panel A is depicted at wFIC = 1. When computed on an interval of wFIC, stable branches of this structure form an attracting surface with a lower and an upper sheet corresponding to families of stable FPs (gray surface) and LCs (yellow surface). The upper boundaries of the stable LC surface are given by a family of Hopf bifurcations (blue line) and saddle-node bifurcations of LCs (purple line). The bifurcation diagrams were computed numerically using AUTO-07p [60] and the parameter values given in Table 1.

https://doi.org/10.1371/journal.pcbi.1012595.g003

Notably, wFIC = 1 yields the original bifurcation landscape of the JR model, whereas values wFIC≠1 reshape this structure. To capture this alteration, one can compute vs. I_ext bifurcation diagrams for an interval of wFIC values and obtain surfaces of FPs and LCs, as shown in Fig 3B. Initial conditions and a given point (I_ext, wFIC) in the parameter plane will determine which of the attractors of the JR model the dynamics will approach. At the same time, and will approach the corresponding temporal average of that attractor. This attractor, be it a FP or a LC, will not satisfy , unless (I_ext, wFIC) is chosen appropriately. In other words, the average activity is usually lower or higher than the desired target.

In the case of a small η>0, wFIC is a subject to a slow drift, proportional to . An increase of wFIC will cause a decrease in and vice versa. Therefore, if , wFIC slowly drift towards larger values, causing a decrease of , whereas for , wFIC drifts to smaller value, causing an increase of . Eventually, this leads to , the drift of wFIC stops–it has reached equilibrium. This behavior can be understood analytically. From the representation in Fig 3B, one can assume that and are functions of wFIC. For the dFIC equilibrium we follow that: (7) (8)

Note that holds in the parameter intervals considered in this work. A linear stability analysis at the above equilibrium yields . Therefore, if is fulfilled at the dFIC equilibrium, it will be a stable equilibrium. Given the JR model with the implementation of dFIC as an inhibitory plasticity mechanism, this condition—by construction—is always fulfilled. To sum up, a dFIC equilibrium exists at a given I_ext if a stable FP or LC with at this exact I_ext exists and it follows automatically that this equilibrium is stable.

In the following, we provide simulation results to support the above analysis, using three representative examples shown in Fig 4. They also give insight into the functioning of dFIC and allow to relate wFIC dynamics to the JR bifurcation structure. In the first case, shown in Fig 4A1, 4B1 and 4C1, we set I_ext = 0.2, for which the original JR model only possesses stable LCs with activity levels beyond the desired . Consequently, in the absence of dFIC, the system converges to this LC. The trajectory with active dFIC however, exhibits a drift in wFIC, forcing the activity level to the desired target, which is a fixed point of the JR model. The second case in Fig 4A2, 4B2 and 4C2, on the other hand aims to reach a LC at a target of . In the absence of dFIC, the system goes beyond this target and converges to a fixed point on the upper FP sheet. With dFIC, the activity is reduced and forced onto a fast LC of the JR model. In the last case, Fig 4A3, 4B3 and 4C3, the activity is increased from the fast LC to a larger level, corresponding to a fixed point on the upper sheet.

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Fig 4. Comparison of no-FIC (η = 0) and dynamic FIC (η>0) of a single Jansen-Rit node.

A, B: and wFIC(t) vs. time t, respectively, obtained by direct simulation of Eqs (1), (2) and Eq (4). Each column corresponds to different values of and I_ext. C: Bifurcation structure of the uncoupled Jansen-Rit model with averaging in space. In C, is marked by a red plane and the intersection of this plane with stable equilibria or limit cycles is shown as a red line and marks dFIC equilibria. Results obtained from direct simulation of the no-FIC (η = 0) and dFIC (η>0) are superimposed. Circles mark the start and asterisks the end of these trajectories. The no-FIC and dFIC initial conditions are shifted slightly for visual clarity. The parameter values for the direct simulations are τ_d = 400 ms and , other parameter values are given in Table 1. The bifurcation diagrams were computed numerically using AUTO-07p [60].

https://doi.org/10.1371/journal.pcbi.1012595.g004

Having dFIC equilibria over a wide span of I_ext values is desirable: first, the external current is often a free parameter in neural mass models and second, I_ext will be dynamic and subject to fluctuations in a network setup, as we discuss in the next section. Hence, we look for intersections of attractors, depicted in Figs 3B and 4 with the plane . These intersections are shown in Fig 4 as red lines and represent the set of (I_ext, wFIC) pairs, for which and dFIC is in equilibrium. For the examples in Fig 4, this set is a continuous line. However, the surface of attractors exhibits gaps in the interval . Hence, targets in this interval might lead to a corresponding gap in the set of (I_ext, wFIC) values for which dFIC equilibria exist. In other words, dFIC can fail for targets in this range if μ is not chosen appropriately. Outside on the other hand, dFIC will function as intended and control the system towards the desired target, independent of I_ext. In summary, the correct application of the proposed homeodynamic inhibitory plasticity mechanism for a single JR node requires the existence of an attractor located at the desired value (for LCs, this location is defined through their center), for a given constant I_ext. The attractor does not need to be present at this location for the original JR model (wFIC = 1) but can be found through variation of wFIC.

dFIC in a brain network model

Moving on to the network case, I_ext,i now contains the network input, is time-dependent and spans over a range of values, which is determined by the network activity and indegree of each node i. At the same time, wFIC is now a vector with N_ROIs entries wFIC_i, one for each brain region. An analysis as above is considerably more intricate and beyond the scope of this work. Nevertheless, we want to point out how the above requirement for successful control through dFIC can be translated to a network setup, by making some assumptions. If the network connections are not strong enough to induce fast transients or overly synchronized oscillations, I_ext,i can be separated into three components, (9) namely the constant offset μ, a slowly varying indegree-dependent mean-field input d_iR and small fluctuations ξ_i, which arise from the network and are distinct to each node. If we neglect the last contribution, the constant μ remains and shifts the entire network towards larger external currents. The mean-field input d_iR on the other hand introduces a spread of the network along I_ext, depending on the indegree distribution. Given that R(t) is subject to only slow changes, the above slow-fast analysis for the single JR model holds valid, since R can be treated in the limit case, just as wFIC before. Hence, the behavior of each node in the network can be approximated by the single-node bifurcation structure present in the plane (I_ext, wFIC). The major difference to the single-node case is the distribution of nodes along I_ext. As shown for the single node, outside the interval [0.02, 0.095] of the parameter, i.e., the gap in the attractor surfaces (Figs 3B and 4), we find a continuum of I_ext values, for which dFIC convergence is guaranteed. In the network case and with the assumption of weak coupling, this continuum of I_ext will allow the individual node to approach the dFIC equilibrium, no matter its precise location on the I_ext-axis. Within the gap on the other hand, there can be nodes with I_ext values for which no dFIC equilibrium exists. In other words, in the case of the JR model, dFIC equilibrium on the network level can fail for

In the upcoming sections, we will provide numerical evidence for the functioning of dFIC in different network setups. We assume the success criterion of tuning parameter convergence to be 1% of the distance of the mean of the last L = 5 seconds of the y₀ time-series from the . Before convergence of wFIC_i one can encounter damped oscillations towards the dFIC equilibrium. Hence, we choose initial conditions based on the bifurcation structure to reduce these transients and obtain faster convergence of wFIC_i. When the tuning was successful, we averaged wFIC_i over last L seconds to obtain a single pFIC_i parameter per node in the brain network. We present the simulation results in two separate parts, one for a small network, used to explore the dFIC performance for different values of the time constant and learning rate parameters, τ_d and η, respectively, and then, for a biologically realistic whole-brain network.

Small network simulations

The results of η and τ_d parameter explorations demonstrate the convergence of the tuning algorithm (Fig 5) for for all and τ_d values in the cases of η = 0.01 and 0.005. This means that all single-node wFIC_i state variables converged asymptotically to a stable value necessary for the system to exhibit the desired activity level with the accuracy of 1%. The tuning to a was successful across all conditions except for short τ_d = 10ms and a few η values for G = 10, G = 25, and G = 100 as depicted on Fig 5. Based on these results, τ_d = 1000, η = 0.005 were chosen as parameters for further for simulations as they proved to be robust across both and G values. We have also tested a faster solution with τ_d = 100, η = 0.0025, which, as expected, yielded almost identical results across both and all G values. Fig 6 demonstrates the resulting pFIC_i values as functions of G and indegree, showing a monotonously increasing—albeit nonlinear—relation between the respective values.

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Fig 5. Analysis of η and time window (τ_d) parameter explorations for (A) and (B) under different G values (columns).

Successful tuning (y₀ mean within 1% of ) is marked by yellow crosses. The colors of the squares indicate the percentage of the distance of the y0 state variable activity from the value after tuning. In the two cases of = 0.01 and = 0.103, the evolution of the wFIC variable was on the correct trajectory but the tuning required a longer simulation to converge.

https://doi.org/10.1371/journal.pcbi.1012595.g005

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Fig 6.

Small-network differences in inhibitory control pFIC parameters per node obtained from the tuning procedure for different values (A: = 0.01, B: = 0.103) and G values with the color intensity marking the indegree of a given node. Plot shows the relationship between G values, pFIC values with color-coded indegrees of the nodes of the small connectome. Two different target values were chosen to correspond to sub-bistable and super-bistable JR configurations. Due to the additive impact of noise in post-FIC simulations we chose the = 0.01 further to the left from the saddle node bifurcation and = 0.103 directly at the edge of saddle node bifurcation between the LCs (Fig 2A).

https://doi.org/10.1371/journal.pcbi.1012595.g006

Afterwards, we run the post-FIC simulations to assess the impact of tuning in the small network. The time-series coming from simulations with G = 1, 10, 25 demonstrate the change in activity levels caused by the introduction of the pFIC parameter (Fig 7). It becomes evident that noise and network effects cause an offset between the long-term average activity level and the for two reasons: (a) predominantly—due to stochastic switches between attractors, which can lead to a net-positive (negative) effect for the sub-bistable (super-bistable) case (respectively), and (b) due to the (sigmoidal) nonlinearity of the model’s activation function, which leads always to a net-positive offset, even in the absence of any switches. As a reminder, here we consider sub-bistable targets to be and super-bistable .

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Fig 7.

Average post-FIC y0 state variable for in blue (panels A-C) and (panels D-F) in red for different G values compared against the no-FIC scenario in black for the small network PSPs exhibit bistability between the fixed point (FP) and limit cycles (LCs). The post-FIC potentials are plotted in blue/red solid lines and no-FIC potentials in black dashed lines. Effect of tuning is visible for G > 1. For G = 10, the deviation from the target is visible, which can be explained by the noise induced switching between two dynamical regimes of the system. Note that for = 0.01 the y₀ average is higher than the target due to switches between FP and slow LC and for the the effect is opposite, due to switches between dominant fast LC and slow LC.

https://doi.org/10.1371/journal.pcbi.1012595.g007

Whole-brain post-FIC results

In the second part of the results, we focus on the comparison between post-FIC and no-FIC simulations on the whole-brain network. The resulting PSP time-series show the desired effect of tuning, which, as expected, restricts the baseline activity of each of the nodes to the value. The difference between the and the mean post-FIC y₀ values is caused by the introduction of noise as well as the ability of noise to cause nodes to periodically jump between the regimes of the JR model. The time-series of the state variable y₀ show a predictable deviation from the target activity level, proportional to both the sum of weights and the chosen G parameter value, further highlighting the fact that the dFIC for = 0.007 and places the system in the proximity of multi-stability. The switches between different regimes are less visible for the respective no-FIC simulations (S3 Fig), a phenomenon further investigated in the next section.

We have computed and compared power spectra for the best fitting post-FIC simulations (using the Welch method), and best fitting no-FIC simulations. The power spectra analysis shows a significant post-FIC increase in power in the respective ⍺ and θ frequency bands when compared with the corresponding no-FIC simulations (Fig 8). Increases for occur predominantly in the θ band as the tuning target places the system near the low FP (Fig 8A), where network fluctuations and noise move the system towards the bistable regime between the LCs (to the right on the bifurcation diagram). The presence of multiple peaks in the power spectra, corresponding mainly to θ and slow ⍺ rhythms (<10 Hz), suggests that it is bistable switching and not the slow LC that explains the increased theta power. Similarly, the is a critical point on the fast LC exhibiting alpha oscillations (Fig 8B), where changes in input and noise cause the periodical switches between both alpha and theta LCs. We conclude that the dFIC mechanism is responsible for the increase in power in the respective rhythms. Further support of this claim comes from the Poincaré map analysis.

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Fig 8. The power spectra of all nodes for the two best post-FIC simulations for two different fit measures values (panel A: FC-fit and panel B: MMF fit, blue) and two best fitting no-FIC simulations (red).

https://doi.org/10.1371/journal.pcbi.1012595.g008

dFIC Improves the fit to empirical data

The results of the fitting based on R_FC are shown in Fig 9 and example matrices are available in the S4 Fig. The results indicate that dFIC improves fits between the simulated and empirical FC data as long as allows the system to reach multiple dynamical regimes. When all nodes are forced by the pFIC_i parameter to remain on the fast LC and the repertoire of possible behaviors is significantly diminished. We assumed that the 30 min simulation for BOLD signals approximates well the stationary value of the FC statistic for a given parameter set (including the amplitude of noise).

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Fig 9. Results of functional connectivity fitting.

A-B. The heatmap with the resulting fit measure for all post-FIC and the best fitting simulations without FIC (no-FIC). The red dots mark the best-fitting G parameter per , white square marks the best fitting simulation overall. C-D. Plots C-D illustrate the direct comparison of Pearson-R values for best fitting simulation per C: for sub-bistable and D for super-bistable target values. Max R.: synchronization-adjusted Person correlation coefficient between simulated and empirical data.

https://doi.org/10.1371/journal.pcbi.1012595.g009

FCD fitting results show an overall improvement of fit for most sub-bistable targets and significantly higher best similarity score (1−KSD_FCD) for post-FIC simulations compared with no-FIC settings (Fig 10).

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Fig 10. Heatmap of Kolmogorov-Smirnov-Distance-based similarity measure (1-KSD) for both post-FIC (A) and no-FIC (B) simulations.

The optima are in a very similar range, but they differ slightly suggesting that this measure captures different (dynamical) properties of the BOLD dynamics.

https://doi.org/10.1371/journal.pcbi.1012595.g010

We have combined the results from synchrony-adjusted FC and FCD fitting into a one Multimodal Factor fitness function (MMF). The resulting optima have largely remained similar, but the global optimum shifted to , G = 11 (Fig 11). Interestingly, the FCD distribution of this simulation visually fits the empirical FCD better than the best qualitative 1−KSD_FCD fit (Fig 12B compared with Fig 12C) as the best 1−KSD_FCD only fit contained a second peak.

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Fig 11. Heatmap of Multimodal Factor fit based for both post-FIC (top) and no-FIC (bottom) simulations.

The result that MMF maximizes not directly at the critical point of saddle-node bifurcation ( but at can be explained by two factors: (1) we sampled the space more densely for the sub-bistable regimes than for the super-bistable and (2) ones, as we suspected this range to be optimal, second: the network effects and noise have primarily additive effect, which causes a shift in effective network average of I_ext,i to the right of the value.

https://doi.org/10.1371/journal.pcbi.1012595.g011

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Fig 12. Example of FCD matrices and their distributions.

A: shows the FCD data for the best fitting post-FIC simulation chosen solely based on the Pearson correlation with empirical FC. B: for the generic empirical FCD data, C: for the combined MMF fit, D. shows the FCD data for the best fitting no-FIC simulation based on the combined 1−KSD_FCD fit. On all the plots average empirical FCD distribution is plotted in black. The FCD matrices were threshold at 96^th percentile for clarity.

https://doi.org/10.1371/journal.pcbi.1012595.g012

The plot of all the MMF fits in Fig 13 shows the wide range of good fits for a wide range of sub-bistable and G values and for super-bistable . The difference between the wider sub-bistable range and a single super-bistable value can be attributed to two factors: first, the denser sampling of the former parameter space and the noise, since the offset it induces via network effects is mainly net-positive for the sub-bistable regime due to switches from the FP to the LC.

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Fig 13. Comparison of best fitting results for different .

A: Sub-bistable targets. B: Super-bistable targets. These plots show the viability of the Multimodal Factor (MMF) fitting approach and the positive effect of the dFIC on the range of sub-bistable cases and directly super-bistable case of .

https://doi.org/10.1371/journal.pcbi.1012595.g013

To test if differential MMF fitting results between the post-FIC and no-FIC simulations are statistically significant, we conducted a permutation statistical analysis by sampling without replacement from the same set of MMF values computed for the above shorter time windows, but this time randomly mixing the two conditions of post-FIC and no-FIC simulations. Statistical analysis showed that the difference between post-FIC and no-FIC simulations was significant at a level of p < 10⁻⁶ for almost all parameter combinations. Details of both procedures are further described in the S1 Appendix.

dFIC intensifies “switching” between dynamical regimes

To investigate the switching behavior of the model, we computed Poincaré maps of synaptic potential time-series. This step allowed us to highlight and quantify the effect of the dFIC on the model’s underlying activity. First, we have identified a threshold of activity (c = 6 mV) which can be applied to the resulting maps to identify the regime of the JR model being exhibited by any node at any given time. More in detail, the lower activity FP of JR in presence of noise, yields points (PSP_i,k, PSP_i,k+1) in [0 mV, 6 mV]×[0 mV, 6 mV], the fast LC in [6 mV, 18 mV]×[6 mV, 18 mV], whereas one period of the slow LC will have one point in [0 mV, 6 mV]×[6 mV, 18 mV] and one in [6 mV, 0 mV]×[0 mV, 6 mV], due to the formation of two maxima within one period. If a node traverses multiple regimes however, the point cloud of a single node is going to fulfill multiple of those conditions, allowing to identify which regimes were visited and therefore which dynamics exhibited. By analysis of the pattern formed by individual nodes over time and the pattern of all nodes together, it is possible to extend the understanding of the overall behavior of the model. Further we have calculated how many nodes exhibit behaviors in one, two or three different regimes for more than 7,5% of the simulation time T_p, as a measure of complexity of the resulting signals.

For most of targets in the proximity to critical points of the JR model (), Poincaré analysis shows higher variability of exhibited regimes. That means, more regimes can be reached by any single node and a higher number of nodes are ‘visiting’ a higher number of regimes in post-FIC simulations compared to their no-FIC counterparts. The dFIC algorithm for those targets increases in overall switching behavior for a high number of tuning targets. In contrast for some low G values and ∈ {0.004, 0.007, 0.01} dFIC is binding the activity of the nodes to a single regime. Expectedly, for and , dFIC does not significantly impact the switching behavior. Fig 14 shows the and G values for which FIC caused an increase in the number of showcased behaviors. We demonstrate this effect based on two examples: Fig 15 shows best fitting no-FIC () compared with best fitting post-FIC (), where tuning increases this number significantly (Fig 15A1 and 15A2). This effect can also be observed when the is super-bistable, in which case the no-FIC method is compared with the post-FIC one, both with (Fig 15B1 and 15B2).

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Fig 14. Summary heatmap of increases in number of regimes traversed by the nodes.

If the number of regimes of post-FIC simulations is greater than for no-FIC simulations the map is marked in yellow, in the opposite case in black, and in yellow if there is no difference. The tuning targets that correspond to a single attractor, with large distance to any bistability show lack of improvement with dFIC or decrease in number of nodes in multiple exhibited regimes. In contrast the targets where the system is tuned to a sub-bistable regime and directly super-bistable regime show increases in that measure. Inc: increased, Dec: decreased, NC: no change.

https://doi.org/10.1371/journal.pcbi.1012595.g014

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Fig 15. Poincaré maps for the best no-FIC (A1) and post-FIC (B1) MMF fit.

Each point (PSP_i,k, PSP_i,k+1) maps one PSP maximum PSP_i,k of an individual node i to the next one PSP_i,k+1. The node indegrees are color-coded (low indegree: dark blue, high indegree: yellow). The set threshold at c = 6 mV consistently (visually) separates the FPs and LCs of the JR model. Bar plots at the bottom right corner of each subplot show the percentage of nodes exhibiting 1,2 or 3 regimes during at least 7,5% of the time of the simulation, showing the effect of dFIC. The A2 and B2 panels compare no-FIC with post-FIC simulations for super-bistable settings (). The node indegrees are color-coded low indegree: dark blue, high indegree: yellow. The set threshold at c = 6 mV consistently (visually) separates the FPs and LCs of the JR model. Both plots demonstrate a difference in number of regimes traversed by each (map) of the nodes during the simulation. A decrease in number of nodes in a single regime or an increase in the number of nodes in two or three regimes is considered a richer repertoire of behaviors as a result of implementing dFIC.

https://doi.org/10.1371/journal.pcbi.1012595.g015

Overall, Poincaré mapping analysis for the best fitting simulations according to the MMF showed the positive impact of dFIC on the number of nodes traversing 2 and 3 regimes (Fig 14). The overlap between high values of fits of R_FC, 1−KSD_FCD, and MMF in the parameter plane and the results of our Poincaré mapping analysis suggest a strong link between the overall quality of fits between the simulated and empirical BOLD data and the switching behavior in the PSP time-series. This effect is visible while comparing the heatmaps (Fig 16), suggesting a similar range of potentially optimal JR and dFIC parameters. Our results provide further evidence that in simulations with JR model the switches between different regimes leading to different amplitudes and frequencies of oscillations in the underlying PSP data result in more variable and hence more biologically plausible BOLD signals and BOLD-derived FC matrices.

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Fig 16. The heatmaps showing FC, FCD, MMF fits and the heatmap of increase switching behavior in underlying PSP activity as shown by Poincare analysis.

Heatmaps for all 3 fitting measures (A-C) show a similar range of good fit parameter values spanning from to 0.1. and for wide range of G values. Interestingly a very similar range shows an increased number of switches during post-FIC simulations in PSPs (D).

https://doi.org/10.1371/journal.pcbi.1012595.g016

dFIC, E-I balance and criticality

The balance of excitation and inhibition, postulated as a central property of a healthy neural system, has had a few explicit implementations in research dedicated to brain simulations [8,46,51,75]. Studies show the significance of homeodynamic plasticity in the context of maintaining E-I balance and simulating healthy and more realistic neuronal dynamics, but the connection to the notion of criticality remains debated [25,76,77]. Both concepts have been argued to require some sort of plasticity mechanism [33,36,49,78,79]. It is worth noting that several mechanisms collectively contribute to the brain’s ability to self-regulate E-I balance through synaptic plasticity [80,81]. Recent studies suggest the roles of short-term [82] and long-term synaptic plasticity [83], and structural plasticity [84] in maintaining this balance. Additionally, there are cellular mechanisms distinct from plasticity, but with capability of self-regulation. As an example, spike-frequency adaption, i.e., the property of certain types neurons to reduce their firing frequency upon sustained input, could effectively act as a negative feedback similar to feedback inhibition [85]. Overall, mechanisms like dFIC are implementations of generalised inhibitory synaptic plasticity mechanisms in the BNMs. However, their effectiveness in maintaining E-I balance in simulated networks does not allow for determining the specific biological underpinnings of such mechanisms.

Even though, the E-I balance does not necessarily entail criticality in the context of BNM [86], there has been evidence suggesting that E-I balance, and optimal information processing based on criticality are two tightly related phenomena [24,31,87,88]. For example, researchers have postulated that E-I balance is directly responsible for bringing the cortical activity to criticality in a rat’s visual cortex in vivo [89]. Their research strongly suggests that homeostatic plasticity, linked to E-I balance in general, is responsible for bringing the cortex to the critical state. A recent review on inhibitory plasticity mechanisms points out that E-I balance in neural networks enables complex, non-linear collective behavior [75]. Furthermore, E-I balance can be a necessary condition for more complex dynamics on a whole network scale. Proponents of E-I balance have argued for the role of inhibitory plasticity mechanism in keeping the BNM at criticality in the Deco-Wang model and Wilson-Cowan model, respectively [18,49]. They have provided evidence for the role of such a process suggesting that the brain uses such mechanisms to stay in a state that prevents runaway activity such as epileptic seizures and enriches the dynamics (thanks to proximity to criticality), while staying relatively immune to the strength of noise or global coupling. Interestingly, their results, similar to ours, showed improved correspondence between simulated and empirical functional resting-state fMRI connectivity compared to models without inhibitory plasticity mechanism.

Our findings support the idea that E-I balance plays an important role in maintaining rich and biologically plausible dynamics in BNMs. Depending on the , dFIC positively affects the desired characteristics of simulated neural signals such as variability and richness of behavior. Here, we have provided experimental arguments for choosing the sub-bistable or directly super-bistable targets in the context of the JR model. However, our solution delineated limits and conditions for a wider range of tuning targets including the low-activity or high-activity states, which in future research can be valid tuning targets depending on the specific research question.

The mechanisms related to inhibitory control, similar to dFIC, have potential relevance to modeling E-I balance also in the clinical context. Parameters responsible for the balance between excitation and inhibition have been linked to abnormalities in Alzheimer’s Disease, such as synaptic degeneration, hyperexcitation, white matter atrophy, as well as amyloid-beta and tau levels. In previous studies, inhibitory parameters of BNM were fit to empirical functional data of patients [9,90]. Recently, the JR model has been used to derive biomarkers related to inhibitory control, emphasizing the potential of model-based approaches in the early diagnosis of dementia and Alzheimer’s disease [57,91]. Similarly, the role of inhibitory control in the progression of dementia has been studied using the Wilson-Cowan model [92] and the Deco’s model from 2018 [93,94]. As we have shown, dFIC improves the overall model fitting, facilitating the study of how different dynamical regimes and transitions between them may be linked to various pathological conditions. Additionally, the resulting inhibitory plasticity parameters, can be interpreted as measures of dynamical differences in clinical setting.

To sum up, dFIC is designed to maintain the activity of all nodes at a certain level, by adjusting local inhibition and, in that manner, maintain E-I balance at both local and—indirectly—at the global scale regardless of the chosen target. Our exploration of different activity targets suggests that tuning the system to the proximity of bistability results in increased switching between regimes and better fits to the empirical data. dFIC provides improvement in characteristics observed in the empirical signals. Previous works have treated FIC as an iterative process of simulating, computing activity levels, adjusting weights and then simulating again [8,16,17]. In this work, we embed dynamic FIC as an extension into the JR model, similar to Santos et al. in Wilson-Cowan model [51]. The dFIC equations lead to a slow, asymptotic convergence towards a target average activity of the JR nodes in the whole-brain network model, under conditions relatively easy to be met in most cases (of neural mass models and target dynamical regimes thereof). The analytical treatment of the single-node tuning dynamics reveals the conditions under which convergence towards a target average activity can be guaranteed and provides guidelines for employing dFIC in practice.

Limitations

We have implemented the dFIC mechanism in the JR model, because it is widely used in BNM studies as well as because it is complex enough to allow for multiple dynamical regimes and possible bifurcations among them. However, our results are applicable to other models which have a control quantity (here ) which monotonously decreases with an increasing control parameter (here wFIC). One can assume that a dFIC mechanism, potentially implemented in other neural mass models, would possess this property, since increased inhibition should naturally lead to reduced activity. Beyond this, the stability of the dFIC dynamics only depends on the presence of stable fixed points and LCs (in the uncontrolled model), which are located at the desired target activity. Further analysis is needed to assess how dFIC would change a system’s behavior which possesses chaotic attractors. Therefore, applying dFIC to other neural mass models should be straightforward, a bifurcation analysis being a prerequisite. Future work can take up this task and provide examples of dFIC application to other neural mass models implemented in TVB and frequently used in BNM studies.

In general, although our results assume deterministic dynamics during the dFIC tuning process, we expect them to remain valid in the presence of relatively low additive, white noise, which is generally used in stochastic simulations of BNMs, as well as for our post-FIC simulations. Such a stochastic version of dFIC could also resolve the problem of a possible offset between the and the average y₀ value resulting from stochastic post-FIC simulations, which was observed in our study. Alternatively, tuning in presence of noise could be achieved by running a short stochastic dFIC simulation right after the deterministic dFIC has converged, by using the result of the latter as the initial condition of the former.

Our work utilized fitting static and dynamic characteristics of the simulated BOLD signals to assess their ‘biological plausibility’, including FC, FCD and synchrony. However, it did not extend to the high-temporal resolution signals such as MEG and EEG. Even though, we presented the basic properties of the generated PSP time-series (with and without dFIC), in the frequency domain and via Poincare Maps (Figs 8,14 and 15), we did not extend our fitting procedures to include those other signal modalities. Such multimodal fitting can help in the future determine the relationship between different modalities and further improve both the fitting methods and the homeodynamic fitting procedures.

Further research can also investigate dFIC as a homeodynamic mechanism, possibly relying on inhibitory plasticity, which maintains the state of the brain within limits and responds to external perturbations induced by e.g., cognitive stimuli. This kind of slow timescale modulation of brain dynamics could be of great interest in understanding data, e.g., on FCD dynamics of BOLD signals. In this context it is important to note that we based our simulation on average empirical structural and functional data: average SC, average FC, and average distribution of FCDs were derived from a large HCP dataset. Hence, when considering data of individual subjects, dFIC might yield slightly different results. The overall findings suggesting sub-bistable and directly super-bistable targets as optimal should in the future be tested using the individual subject data, both for normative and clinical cases.