Discovering governing equations that describe complex chaotic systems remains a fundamental challenge in physics and neuroscience. Here, we introduce a method, the PEM-UDE, that combines the prediction-error method with universal differential equations to extract interpretable mathematical expressions from chaotic dynamical systems, even with limited or noisy observations. This approach succeeds where traditional techniques fail by smoothing optimization landscapes and removing the chaotic properties during the fitting process without distorting optimal parameters. We demonstrate its efficacy by recovering hidden states in the Rössler system and reconstructing dynamics from noise-corrupted electrical circuit data, where the correct functional form of the dynamics are recovered even when one of the observed time series is corrupted by noise 5x the magnitude of the true signal. We showcase that this method with prior knowledge embeddings is able to recover the correct dynamics where direct symbolic regression methods, such as STLSQ, fail with the given amount of data and noise. Importantly, when applied to neural populations, our method derives novel governing equations that respect biological constraints such as network sparsity – a constraint necessary for cortical information processing yet not captured in next-generation neural mass models – while preserving microscale neuronal parameters. These equations predict an emergent relationship between connection density and both oscillation frequency and synchrony in neural circuits. We validate these predictions using three intracranial electrode recording datasets from the medial entorhinal cortex, prefrontal cortex, and orbitofrontal cortex. Our work provides a pathway to develop mechanistic, multi-scale brain models that generalize across diverse neural architectures, bridging the gap between single-neuron dynamics and macroscale brain activity.

2.1.1 Overview

2.1.2 UDE approach

2.1.3 PEM approach

2.1.4 Combining PEM-UDE into single method

2.1.5 Symbolic recovery of terms from PEM-trained UDE

We begin by demonstrating the utility of the PEM-UDE approach in fitting the Rössler system (Fig. 2). This system was chosen as it is a simple set of ODEs with only a single nonlinear term and yet has a rich set of chaotic dynamics for the right parameter set [22]. In this demonstration, we assume we know the functional form of \( x\) and \( z\) but have no information about the functional form of \( y\) , and so must learn the dynamics from the time series observations. With only this challenge, a traditional UDE fails to fit well (Fig. 2a), with the UDE-learned dynamics typically converging to the mean of the time series (typically a local minimum in the objective function). By contrast, the PEM-UDE approach is able to converge successfully after training, with a clear difference in performance even after the first 10% of the total training time (at which point the standard UDE has already neared its local minimum).

The reason for this performance increase lies in the way that the PEM approach alters the landscape over which the UDE is being trained (Fig. 2b). We note that the unmodified landscape is extremely rough and contains several local minima, with only a very sharp decrease in loss at the true global minimum (\( a=0.2\) in this case). By adding the PEM to the dynamics of the system during training of the UDE, the landscape becomes much more approachable via standard optimization techniques. Further, the choice of \( K\) can be made to directly tune the steepness of the landscape. This should be done in conjunction with considering how the PEM will smooth out the landscapes of other parameters in states where the PEM is not applied. For reasonable choices of \( K\) , the addition of PEM will not substantially deviate the fit dynamics from the true minima along other parameters (explored more fully in Fig. 6). Importantly, as higher levels of filtering are introduced with larger \( K\) , finding the true functional form of the entire system will rely more upon the accuracy of the estimates of the un-fit parameters (Fig. 7). When using the PEM-UDE combination approach, it is therefore best to select a hyperparameter \( K\) that provides a smooth enough landscape for optimization to occur but is steep enough that the global minimum is still clearly distinguishable from the surrounding area to allow for noisier estimates of unmeasured parameters.

At its core, the PEM-UDE approach works because the prediction error correction removes the sensitive dependence on parameters from the system being fit, effectively negating the hallmark trait of chaotic systems. As shown in Fig. 9a, when \( K\) is small there is an \( \mathcal{O}(1)\) divergence with respect to any deviation in the initial parameter choice (illustrated as a deviation of 1e-9 in Fig. 9a), but as \( K\) is sufficiently large (\( K\approx0.18\) in this example) this sensitive dependence is eliminated and the error between the two trajectories shrinks. For sufficiently large \( K\) , this error will depend on the size of the deviation in the parameter, and \( K\) can be tuned based on the estimate of this deviation (Fig. 9b). This property, along with the fact that when the error term in Eq. 6a shrinks to zero the dynamics of the original system are fit, means that the PEM perturbed system is well-behaved (i.e. non-chaotic) and has a loss minimum at the same point as the original system. As such, the L2 norm trajectory fitting approach which is unstable on the original chaotic system is thus stabilized on the PEM-UDE, allowing the fitting process to convergence while leading to an effectively unbiased final fit.

The dynamics learned by the PEM-UDE fit the system well, while the initial estimate of the UDE alone does not offer an accurate prediction even within the Lyapunov time of the system (Fig. 2c). For the parameters selected here, the Lyapunov time of the system is \( \lambda^{-1} \approx 10\) , with most nearby areas of parameter space having similar characteristic times (\( \lambda^{-1} < 25\) ; see Methods for more details). While the mean error of the UDE alone continues to grow throughout the simulation time, the mean error at each timestep of the PEM-UDE fit dynamics is 2-3 orders of magnitude smaller than the dynamics themselves.

To generalize these dynamics beyond the original observations, we extracted a symbolic form of the equations using STLSQ fit to the dynamics of the PEM-UDE system and the outputs of the UDE at each timestep. The functional form of the missing state is directly recovered by STLSQ (\( y = p_1x + p_2y\) ), with the initial fits of parameters close to the true values (\( p_1 = 1.01\) , \( p_2=0.17\) , while the true values are \( p_1=1\) , \( p_2=0.2\) ). By adding an additional optimization step to tune the parameters of this learned symbolic equation after STLSQ, we are able to converge to the true values (accurate to three decimal places), which then generalize the dynamics forward in time beyond the original simulations (Fig. 2d). Importantly, this pipeline provides an analytical form of the dynamics that not only generalizes forwards in time but can also be interrogated for insights into the physical system.

We next present a second example of the PEM-UDE method being fit to a chaotic system, with two important differences compared to the Rössler system. First, this example is one of a physically realized electrical circuit, with the circuit diagram shown in Fig. 3a, and the parameters for the physical circuit (as described in [53]) given in 2. Conceptually, this offers a transition from the idealized chaotic dynamics of the previous system to more real-world applications, as observing chaos in electrical circuits has practical implications [54]. To extend our method further, we create a second challenge – adding a noisy observation to one of the circuit components (\( V_1\) , denoted by the red box shown in Fig. 3a). The true equations for this system are given in 13.a, with the dimensionless form used in this example given by 14.a. In this section, we assume there is an unknown faulty observation of the first state \( x\) , such that there is noise around the observation at 5x the magnitude of the true signal (shown in Fig. 3b). This creates a difficulty for the PEM-UDE fit: the true dynamics (Fig. 3c) are now masked by a significant amount of noise that grossly distorts the shape of the attractor being fit (Fig. 3d).

We assume that this noise is unknown at the time of fitting and proceed with fitting the PEM-UDE to the corrupted data using the same approach as above (i.e., fitting the functional form shown in 15.a). Despite the significant observational noise present in one of the states, STLSQ applied to the PEM-UDE fit dynamics recovers the exact functional form of the dynamics (Fig. 3e). However, we note that this approach does not provide the original parameters, although the form is correct. In this approach, however, we note that we have shifted the problem from one of unknown dynamics to one of simple parameter optimization, to which other approaches (e.g., simulation-based inference [55]) can be fruitfully applied. This also gives us confidence that even in cases of extremely noisy observational data, the PEM-UDE approach can recover valuable information about the true dynamics of the system.

This example also provides a useful demonstration of why fitting the symbolic form of a trained UDE outperforms a STLSQ fit of the original dataset. As a comparison to the dynamics shown in Fig. 3e, we fit the same noisy dataset directly using the SINDy approach [10, 56], i.e. calculating a dataset of smoothed derivative estimates and performing the STLSQ symbolic regression method directly on the derivative estimates. As shown in Table 1, no recovery of the true symbolic form is possible with this method across a range of the ADMM hyperparameter \( \lambda\) , with more sparse representations incorrectly dropping the noisy state \( x\) from the expression entirely. Recovering the symbolic form of the trained UDE is more accurate as the UDE provides a noise-free (or greatly noise-reduced) version of the dynamics that can be sampled with arbitrary density, thereby mitigating the sensitivity that symbolic regression methods show to noisy data.

Having demonstrated the utility of this approach in learning forms of chaotic physical systems, we next turn to deriving novel expressions for the mean-field activity of neuron populations. The data needed to study the human brain inherently bridges multiple scales, as individual neuronal activity gives rise to synchronized actions that define interacting functional circuits (Fig. 4a-c). Traditional neural mass models represent the average activity of neuron populations by modeling specific phenomena (e.g., excitatory/inhibitory balance in the Wilson-Cowan model [39], ion gradient dynamics in excitatory populations in the Larter-Breakspear model [41, 57]), but lack a direct tie to many biophysical parameters (e.g., neurotransmitters) that are essential in understanding inter-neuron dynamics. To overcome this, next-generation neural mass models (NGNMMs) take a different approach, providing an analytical derivation of the mean-field dynamics that retains the biophysical details of individual neurons while also capturing the emergent properties at the population level [42, 43]. The derivation and its key assumptions are discussed further in the Methods.

This analytical form of mean-field dynamics offered by NGNMMs provides many advantages over traditional neural mass models. First, there is a direct connection between the single neuron parameters and the dynamics of the neural population, a feature that is only available in certain populations of traditional neural masses tuned to be sensitive to these parameters [41, 57]. Second, NGNMMs capture emergent properties of neural populations that traditional neural masses do not. For example, \( \beta\) -rebound – where the suppressed activity in the \( \beta\) band increases in power after being suppressed (a commonly observed phenomenon in motor cortex studies) – is observed as a feature of coupled NGNMMs that are built only to oscillate in the \( \beta\) band, with no functional constraints that force this activity to occur [44].

Despite these advantages, however, there are some notable limitations to the NGNMM approach, mostly involving the assumptions necessary for the Ott-Antonsen ansatz [45], a key component of the NGNMM derivation. Most important from a biophysical standpoint is the assumption of all-to-all connectivity of the neuron population. This is approximately correct in deep regions (e.g., hippocampus, entorhinal cortex) that have significant within-region collaterals [43]. Immediate neighbors of these regions tend to have fewer collaterals, however, like the thalamostriatal connection patterns [58]. The assumption of full connectivity becomes obviously false in the cortex [48], which typically has very sparse connectivity (on the order of 1-5% of all possible connections; Fig. 4d). Accounting for sparsity in NGNMMs is, therefore, a key problem that must be solved for them to be generalizable in new contexts.

As there is disagreement on the correct analytical approach to adjust the derivation of NGNMMs for sparsity [47, 59, 60], we propose to sidestep the issue of an analytical derivation entirely and simply learn the form of the sparse networks directly from the data with the PEM-UDE method. Following the analytical approach for Izhikevich neurons [43], we begin with a large population of neurons described by Eq. 16 with synaptic dynamics in Eq. 18. In the all-to-all case, the analytical NGNMM describing the activity of this population is given by Eq. 20.a. As we begin to prune connections in the graph to introduce sparsity, however, we see that this form no longer accurately represents the population activity. We therefore learn correction terms for the mean firing rate \( r\) and voltage \( v\) given a reduced probability of connectivity \( p(c)\) , where \( p(c) \in [0.05, 1]\) . Although there are additional states for the recovery current \( w\) and the synaptic dynamics \( s\) , we only learn from the firing rate and voltage states and train only on their data, as these are the data typically available from experiments. Learning directly from these data sources thus allows us to demonstrate the applicability of this approach in learning directly from experimental data.

The PEM-UDE approach is able to capture the mean-field dynamics of the network under different degrees of sparsity to a high degree of accuracy (Fig. 4e). This accuracy does not include the initial conditions; this is discussed further in the limitations and future directions portion of the discussion. It does, however, capture a noticeable shift in the dominant frequency of oscillations present in the network as a function of sparsity (Fig. 4f). Densely connected regions typically have a primary frequency peak in the \( \theta\) -band, while the most sparse networks exhibit a primary peak in the \( \alpha\) -band. This emerges as a natural feature of the learned dynamics and is also observed in experimental data. Hippocampal rhythms are typically in \( \theta\) frequencies and emerge from the densely connected areas of the brain [61], while \( \alpha\) frequencies typically emerge from sparsely connected cortical regions before propagating through the brain [62]. The PEM-UDE learned dynamics therefore preserve and extend the strength of NGNMMs in capturing the emergent properties (e.g., synchrony) at the mean-field scale by accurately captuging the ergodic properties of the system.

As in the previous two systems, we extend the PEM-UDE learned dynamics described in Eq. 22.a by fitting a symbolic form of the dynamics to the UDE network with STLSQ, providing an analytical form of the solution (shown in Eq. 23.a, with fit parameters in Table 4). As NGNMMs are derived based on the assumption of a population of coupled oscillators, there is an exact expression for the degree of synchrony within the population given by the expression for the Kuramoto order parameter \( Z\)

In the case of the symbolic form of the PEM-UDE learned network dynamics, we observe an emergent prediction that more densely connected regions will tend to have higher degrees of synchrony as a baseline state. In comparison, more sparsely connected regions will tend to exhibit lower degrees of synchrony when not acted upon by an external drive (Fig. 4g). This forms an interesting and unexpected prediction from the discovered equations for networks of different sparsity. To test the validity of this prediction, we examined three different datasets of intracranial electrode recordings from regions of different degrees of sparsity (Fig. 5). These datasets included recordings from the rat medial entorhinal cortex (MEC; N=6), a deep region near the hippocampus with high intraregional connectivity [63, 64]; the rat prefrontal cortex (N=10), which has significantly sparser connectivity [65, 66]; and human orbitofrontal cortex (N=10), which exhibits a similarly high degree of sparsity [67, 68]. As predicted by the discovered equations, we note that the deeper, more densely connected structures exhibit a higher degree of synchrony than either of the cortical datasets, and this is a broad trend across individuals, although there is significant variability between subjects. These results suggest that the learned equations are not merely capturing the dynamics on which they are trained but have been able to predict features beyond their initial training data, indicating a more general applicability than the original form allowed.

The Julia ecosystem has a robust infrastructure for scientific machine learning [16] that we use to build the PEM-UDE method described in this work. As setting up and training UDEs in Julia has been discussed in other publications ([16, 7] are notable examples), we encourage interested readers to simply peruse the code to see the details of the implementation and content ourselves with a few brief technical notes here on details about the choices of solvers, network structure, fitting procedure, and symbolic regression.

In this work, we default to non-stiff solvers (typically Vern7), largely for speed considerations. As long as the UDE network is sufficiently large to allow for a decent approximation of the unknown dynamics and the system itself is not incredibly stiff, a non-stiff adaptive solver should provide sufficiently accurate fits to the dynamics. If the system is known or highly suspected to contain a great deal of stiffness, then switching to a stiff solver (e.g., FBDF) is recommended and is easy to do in the provided code, as the Julia ecosystem has a wide range of solvers available. This switch may become necessary either because chaotic systems are often stiff in areas, or even if the system itself is not stiff (as the neural systems discussed can be in the correct parameter space), the inclusion of a universal approximator will often lead to a stiff problem, at least until the initial parameter weights are adjusted to fit the problem. We recommend increasing the size of the universal approximator network before defaulting to a stiff solver, as this can significantly impact performance.

We choose the radial basis function (RBF; notation given in [78]) as the activation function for these layers, given by

The RBF is selected for both theoretical and practical reasons. There is a robust body of literature showing that the RBF is useful in the context of universal approximation [78, 79, 80]. Because a neural network of this form can approximate any function given a sufficient depth of the network [78], the choice of network size is therefore arbitrary beyond a specific size. It can be set freely based on the availability of computational resources for training. For the networks shown in this work, we choose a depth of four layers, where for an input dimension \( N_1\) and output dimension \( N_2\) , each network is of size \( N_1 \times 10 \times 10 \times N_2\) .

As the PEM portion of training requires interpolating the original training data to the solver time step, we use DataInterpolations.jl to do simple linear interpolation between observations [81]. Although there are many optimizers provided to train the UDEs, we have defaulted to using the AdamW method (decoupled weight decay and gradient updates; [82]) in all of the results presented here.

While the dynamics fit by the combined PEM-UDE method appear to capture the true dynamics of the system, it is important to note that they are trained on limited data, and evaluating robustness – particularly in chaotic systems with their sensitivity to initial conditions – is challenging to carry out. The approach to mitigating this is to fit a symbolic expression to the dynamics of the UDE networks that we train to provide a form of the dynamics that can be interrogated with standard analytical approaches. For the purposes of this paper, we do not make any claims as to the best way of extracting the symbolic form of the UDE network and note that there are advantages to different approaches [3]. We provide examples of two of these approaches [10, 14] in the code repository associated with this work. As the STLSQ approach is significantly faster than the genetic algorithm approach when large amounts of data (\( N \gtrapprox 10,000\) data points) are required to fit the functional form accurately [14], we default to this method for the results shown here. In principle, however, the trained UDE can be used to generate a subsample of arbitrary density, so either approach can be used, depending on the requirements of the problem at hand.

The notation for the symbolic recovery of terms from neural networks is adapted from [10], so we briefly explain it here. \( \Theta\) is defined as the feature library with some linear combination of features given by weights \( \xi\) so that \( y=\Theta\xi\) . To reduce the library components that constitute the functional form, a sparse solution is found via LASSO [83]. The objective function is therefore given by

In the implementation used in this work, this objective is approximated by sequential thresholded least-squares (STLSQ) solved for using the alternating direction method of multipliers (ADMM; [84]), as implemented in DataDrivenDiffEq.jl [56].

The Rössler system is a set of three nonlinear differential equations as a simplified model for studying chaotic dynamics, as it contains only one nonlinear term but still generates complex chaotic behavior. The dynamics of the system are described by the 3D ODE system:

For specific parameter values (in the case presented here, \( a=0.2\) , \( b=0.22\) , \( c=14\) ), it produces the distinctive funnel-shaped Rössler attractor that exhibits sensitivity to initial conditions, fractal structure, and period-doubling chaotic activity. We chose this as an initial demonstration of the ability of PEM-UDE combination learning as, despite its simplicity, the Rössler system demonstrates essential chaotic properties including positive Lyapunov exponents, topological entropy, and various bifurcation sequences that have helped researchers understand transitions between ordered and chaotic regimes in dynamical systems. Additionally, the Rössler system has been extended to analytically describe certain physical problems (e.g., mechanical vibrations, atmospheric variability).

In the results presented here, we use a universal approximator \( U\) to learn the dynamics of the entire second state, assisted by the Kalman-filtered observational data through the PEM technique. The system simulated during learning is therefore given by

For the results presented in the main text, we select \( K=0.25\) . This choice is driven by the steepness of the parameter landscapes as discussed in the results, and other reasonable values of this hyperparameter will not affect the accuracy of the learned dynamics, provided an appropriate solver is chosen. The results presented here are based on 100,000 training epochs with the AdamW optimizer [82]. The loss function used to train the UDE network is simply mean-squared error between the predicted states at each time step and the true time series (with a small amount of noise added for robustness during training):

For the parameters selected in our analyses, we characterized the degree of chaos via the Lyapunov time of the system as \( \lambda^{-1} \approx 10\) . Most parameter combinations in the surrounding space have similar values (\( >80\%\) have \( \lambda^{-1} \in [0, 25]\) - see supplemental code for computation of the empirical CDF of the maximum Lyapunov exponents). Lyapunov exponents were computed using standard functions offered in ChaosTools.jl [85, 86].

Petrzela and Polak [72] introduced a category of analog electrical circuits exhibiting chaotic dynamics under three constraints: a canonical form, structurally stable attractors, and novel dynamics. Here, we work with a particular realization that provides chaotic dynamics that are visually interesting and can be physically realized in a simple circuit, with the attractor seen on an oscilloscope (demonstrated in [53], from which we take the parameters and non-dimensional form discussed here). For the circuit shown in 3a, the equations describing the three observed states are given by

For convenience in numerical simulations, we work with the non-dimensional form of the system (i.e., one in which physical constants are replaced with the minimal number of parameters to represent the same dynamics). The non-dimensional form of this system is given by

All parameters for both the dimensional and dimensionless systems are taken from [53], and are provided in Extended Data Table 2.

The form of the equations learned by the PEM-UDE method is given by:

The loss function used to train this PEM-UDE combination is the same as 12 to demonstrate the robustness of the system to a poorly observed state. If it is known a priori that the observations of a state are corrupted to that degree, it could be worth modifying the loss specified in 12 to weight the error in this state less during training or to drop the observations altogether. As shown in the next section, the PEM-UDE training method can recover dynamics even when some states are completely hidden, provided the initial guess of their dynamics is sufficiently accurate.

Next-generation neural mass models rely on the Ott-Antonsen (OA) ansatz to derive the mean-field activity of a population of neurons [45, 42]. Briefly, this approach was introduced as a means of bypassing the need to set up an entire Fokker-Planck equation to derive the mean-field activity of a population of coupled oscillators, provided certain key assumptions are met. The OA ansatz [45] relies on the assumption that the population of coupled oscillators being examined exists in the thermodynamic limit (i.e., \( N \rightarrow \infty\) ) and that the probability distribution of the observable (typically voltage) in phase space converges to a Lorentzian. However, this derivation can approximate other distributions to a reasonable degree of accuracy [47]. The OA ansatz approach also assumes that the noise input into each neuron receives a heterogeneous, quenched noise input \( \eta\) drawn from a general distribution. The key point of this assumption is that the currents cannot be stochastic; otherwise, the prior assumption (the Lorentzian ansatz) is violated, which is further discussed below. The OA ansatz also assumes that the conditional probabilities of the macroscopic variables are separable. This assumption and its limitations have been discussed at length [74], and a second-order circular cumulant approximation has been derived for neural mass models [59]. Finally, the OA ansatz makes the assumption we address in this paper: The network being approximated has an all-to-all connectivity structure. As we have discussed above, this is a fundamentally non-physiological assumption for most regions of the brain, and removing this assumption is an important step in developing more accurate neural mass models.

As we do not require the OA ansatz for the work presented here beyond the initial assumption of the functional form for population activity that will be used to train the UDE, we omit most of the derivation and discuss only the key points. The equations shown here are adapted from earlier work that shows the entire derivation [43]. Following this work, we simulate a network of Izhikevich neurons that take the form

where \( V_j\) and \( w_j\) are the voltage and recovery current, respectively, of neuron \( j \in [1,N]\) for a network of \( N\) neurons. The voltage activity in these neurons depends on the internal neuronal dynamics, the distributed noise source \( n_j\) described above, any external driving current \( I_\text{ext}\) , and the synaptic current from other interneuronal connections \( I_{\text{syn},j}\) . The synaptic activity is given by

Each neuron therefore receives an impulse of \( \frac{s_\text{jump}}{N}\) at every time that a presynaptic neuron \( k\) fires. If the network is structured to fulfill the assumptions discussed above, the mean-field activity of the entire poulation can be derived using the OA ansatz and will take the form:

where \( r\) is the population firing rate, \( v\) is the mean voltage, \( w\) the mean recovery current, and \( s\) the mean synaptic activity. As discussed in earlier work, this is an accurate representation of the mean-field activity for fully connected networks [43].

For all the simulations presented here, we simulate a network of 1,000 Izhikevich neurons with varying degrees of network sparsity, where the probability of two neurons forming a synapse is given by \( p(c) \in[0.05, 0.1, 0.2,...1.0]\) as indicated for the sparsities shown in Fig. 4. In a standard network of neurons, as the sparsity of connections is varied, it is necessary to rescale the synaptic currents proportionally with the median in-degree of connectivity \( K\) to maintain balanced (synchronous) activity. Previous work has shown that rescaling the synaptic conductance with \( g \propto \frac{1}{\sqrt{K}}\) – with similar rescaling of background currents, if present – provides an accurate scaling for a broad spectrum of network arrangements (in-degrees with Lorentzian, Gaussian, and Erdős-Rényi distributions) [47]. Following this work, we rescale synaptic conductance with \( p(c)\) such that

As in the previous sections, we set up the UDE to learn the unknown terms for the states we are interested in, but with two key differences. We are learning terms for both the first and second states in this derivation and thus applying PEM to both of these states as well. We also modify Eq. 12 to train on only data for the first two states, as these are the only states typically observable in experimental data. The initial parameters for this system are given in Table 3, and the formal description of the PEM-UDE system is given by

After learning the dynamics with the PEM-UDE approach and recovering the symbolic form of the UDE expressions via STLSQ, the new NGNMM with correction factors for sparsity is given by the parameter values in Table 4 and by the functional form:

As acquisition and preprocessing of each of the three experimental datasets have been described in detail in their respective publications [63, 65, 67], we only briefly describe the datasets here. The rat MEC dataset [64] was acquired from six male Long-Evans rats, with electrode recordings acquired during a long period of roaming in a circular track. The rat prefrontal cortex dataset [66] was acquired from four male Long-Evans rats during 10 trials in which they successfully learned a behavioral task, including the rest periods before and after the task recordings. The human orbitofrontal cortex dataset [68] was acquired during electrocorticography recordings of 10 neurosurgical patients undergoing evaluation for epilepsy. The patients were asked to play a decision-making game, and the recordings used for this data are the inter-trial rest portions of the ECoG data. For all datasets, the synchrony between electrodes within the region was computed as the phase-locking value (PLV) between the angle of Hilbert-transformed signals (\( \phi\) ) from each electrode [87, 88]. Following prior work [48], spike data from the dataset, which recorded spike times rather than LFPs, were converted to an LFP proxy by windowed spike counts, with the PLV calculation performed on these instead. PLV is given by the formula

for all discretized \( t \in [1, N]\) and all pairs of electrodes \( (i, j)\) .

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