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Simulations Approaching Data: Cortical Slow Waves in Inferred Models of the Whole Hemisphere of Mouse
Cristiano Capone, Chiara De Luca, Giulia De Bonis, Robin Gutzen, Irene Bernava, Elena Pastorelli, Francesco Simula, Cosimo Lupo, Leonardo Tonielli, Anna Letizia Allegra Mascaro, Francesco Resta, Francesco Pavone, Micheal Denker, Pier Stanislao Paolucci
TL;DR
The paper asks how to infer and evaluate high-resolution models of cortex-wide slow-wave dynamics when model parameters cannot be directly compared with biological observables and recordings are limited. It builds a two-step mean-field inference procedure using likelihood optimization, periodic neuromodulation, and direct comparison of wave observables. The resulting simulations reproduce qualitative propagation patterns and share experimentally observed propagation modes, while shuffled simulations produce orthogonal modes.
Problem
Inferred model parameters are difficult to assess biologically, and conventional constraints such as mean activity and spatial correlations do not capture the full diversity of spontaneous traveling-wave dynamics.
Method
The paper builds a high-resolution whole-hemisphere mean-field model through two-step inference, combining likelihood-based parameter optimization with observable-based optimization of periodic neuromodulation.
Results
The simulations reproduce qualitative slow-wave propagation patterns and express the same propagation modes as experimental data, unlike the shuffled control.
Takeaways & Limitations
Flexible macroscopic observables provide a basis for comparing experiments and simulations and validating data-constrained cortical-wave models.
Takeaways & Limitations
The study is based on data acquired under Ketamine-Xylazine anesthesia, limiting its experimental setting.
Abstract
from arXiv · showhide
Thanks to novel, powerful brain activity recording techniques, we can create data-driven models from thousands of recording channels and large portions of the cortex, which can improve our understanding of brain-states neuromodulation and the related richness of traveling waves dynamics. We investigate the inference of data-driven models and the comparison among experiments and simulations, through the characterization of the spatio-temporal features of cortical waves in experimental recordings and simulations. Inference is built in two steps: the inner loop that optimizes by likelihood maximization a mean-field model, and the outer loop that optimizes a periodic neuro-modulation by relying on direct comparison of observables apt for the characterization of cortical slow waves. The model is capable to reproduce most of the features of the non-stationary and non-linear dynamics displayed by the high-resolution recording of the in-vivo mouse brain obtained by wide-field calcium imaging techniques. The proposed approach is of interest for both experimental and computational neuroscientists.
Introduction
The paper develops an inverse, data-constrained modeling strategy for cortical slow waves recorded across the mouse cortex. It addresses limited observability and underdetermination by combining flexible wave observables, anatomical priors, and a two-step inference procedure.
- Introduction: Wide-field calcium imaging captures cortex-wide neural dynamics despite lower temporal sampling and limited single-cell resolution.Its signal-to-noise ratio and spatio-temporal resolution support recordings across brain areas.
- Introduction: High-resolution recordings contain 10.000 pixels per frame at 50 µm×50 µm resolution sampled every 40 ms, challenging models to be descriptive and predictive.
- Introduction: Model validation cannot directly compare inferred internal parameters with biological observables, while average activity and site correlations leave other wave properties weakly constrained.The paper therefore targets observables including slow-wave frequency, propagation speed, direction, and wavefront shapes.
- Introduction: Six 40-second recordings create an underdetermined inference problem, motivating anatomical and neurophysiological priors.The proposed prior decomposes connectivity into short-range lateral and long-range white-matter-mediated contributions, with lateral propagation modeled using local elliptic kernels.
- Introduction: The two-step approach combines experimental knowledge with simulations to identify ingredients associated with cortical-wave modes, including neuromodulation and spatial heterogeneity of connectivity and local parameters.
Results
The study infers a data-constrained cortical model in two stages and evaluates it using local slow-wave observables. Sequential likelihood-based parameter inference and neuro-modulation optimization produce simulations that reproduce qualitative propagation, wave statistics, and propagation modes, while the inner loop alone remains overly regular.
- Data and observables: The model uses wide-field calcium imaging of anesthetized mouse cortex and characterizes slow waves through local velocity, direction, and inter-wave interval observables.Calcium deconvolution provides a proxy for local excitatory-neuron activity before model inference and comparison.
- Two-step inference method: The inner loop estimates spatially varying model parameters by likelihood maximization, with validation likelihood used to monitor over-fitting.The inferred parameters include connectivity, external input, and spike-frequency adaptation, and their spatial maps are smooth.
- Two-step inference method: The inner-loop model generates oscillations and traveling wavefronts but produces substantially more regular down-state durations and wavefront shapes than experiments.Its local-in-time constraints do not capture the long-term temporal richness observed in the recordings.
- Two-step inference method: The outer loop minimizes a combined distance between experimental and simulated distributions of speed, direction, and inter-wave interval while optimizing neuro-modulation amplitude and period.The distance is computed from cumulative distributions using the Earth Mover’s Distance, with simulations rejected when state durations mismatch experimental distributions.
- Results: Sequential application of both loops reduces stereotypization and introduces variability resembling the biological phenomenon.Qualitative frames show comparable wave origins, extensions, activity, and activation patterns between simulations and data.
- Results: GMM analysis identifies the same propagation-mode centroids in data and two-step simulations, whereas shuffled simulations produce modes orthogonal to those found in data.When experimental and corresponding simulated wave events are pooled, the GMM is reported to be “fooled” by the simulation.
Discussion
The paper frames two-step inference as a way to make high-resolution cortical models reproduce experimentally observed dynamics while addressing parameter interpretation and individual connectivity variation. Its mean-field framework and shared analysis tools support comparisons between experimental and simulated slow waves, but the present scope remains anesthetized, spontaneous activity with a limited neuro-modulation form.
- Modeling framework: The method automatically builds a high-resolution mean-field model of the mouse’s whole cortical hemisphere.The model is based on populations of interconnected spiking neurons and is intended for large-scale cortical dynamics.
- Inference and validation: Neuro-modulation reduces stereotyped wave collections and introduces variability that more closely mimics biological dynamics in raster-plot comparisons.Figure 5 compares experimental data with simulations from the inner loop alone and from the outer loop with optimal neuro-modulation.
- Inference and validation: The two-step method adjusts generative-model parameters beyond likelihood maximization because hidden variables can make inferred dynamics unrepresentative of the original system.The method addresses a central risk in neuronal model inference: reproducing fitted observables without reproducing comparable temporal dynamics.
- Inference and validation: GMM analysis identifies four propagation modes in combined experimental and simulated collections, while shuffled simulations segregate from experimental events.The shuffled control provides posterior validation that the optimal simulation’s apparent similarity is not produced by a clustering failure.
- Individualized modeling: Individual connectivity inference complements structural and functional connectivity atlases by retaining brain-specific anatomical and activity-related variation.Structural atlases average across individuals and ex-vivo samples, whereas functional connectivity depends on brain state, arousal, and neuromodulatory influences.
- Analysis framework: The framework develops analysis tools that identify wavefronts and quantify their features in both experimental and simulated data.These tools support direct comparison of cortical-wave observables rather than relying only on internal model parameters.
- Scope and limitations: The present study considers only spontaneous dynamics under Ketamine-Xylazine anesthesia, leaving responses to external inputs for future work.The authors also acknowledge that the chosen neuro-modulation shape may have limited descriptive power and plan more general models.
Methods
The methods combine wide-field calcium imaging with deconvolution and a population-based AdEx mean-field model. Anatomical priors, adaptation, and simplified noise assumptions constrain inference from cortex-wide recordings.
- Experimental data: Six-month-old mice were anesthetized with ketamine and xylazine, and their right hemispheres were recorded using wide-field fluorescence imaging.The dataset used GCaMP6f-labeled excitatory neurons and a chronic transcranial window.
- Experimental data: Each optical pixel was modeled as proportional to local average excitatory activity, while GCaMP6f dynamics were represented by a log-normal response.The calcium indicator’s slow response motivates modeling the measurement process before inference.
- Deconvolution: The signal was deconvolved in Fourier space using the estimated indicator response, with a 6.25 Hz low-pass cutoff.The procedure applies the fast Fourier transform, divides by the response spectrum, and reconstructs the firing-rate estimate with the inverse transform.
- Generative model: The generative model contains approximately 1400 interacting populations, each associated with an imaging pixel and described by its average firing rate.Each population contains neurons coupled through average synaptic weights and connectivity degrees.
- Generative model: Spike-frequency adaptation was included because its strength and recurrent connectivity are associated with slow-oscillation regimes.Adaptation increases after spikes, decays toward rest without spikes, and is modulated through its strength parameter.
- Generative model: A first-order mean-field model used a constant small input variance, while its gain function mapped noisy input statistics to population activity.The constant variance simplifies inference, although a full treatment would require evaluating a two-dimensional function and more computation.
The Inner Loop: likelihood maximization
The inner loop estimates model parameters by maximizing the likelihood of observed activity histories. Independent data chunks use held-out validation periods to reduce overfitting risk.
- Likelihood maximization: Model parameters are inferred by maximizing the log-likelihood of an observed activity history.The optimization requires analytical derivatives of the likelihood with respect to each parameter.
- Optimization: The derivative expression includes a constant c2 that can be absorbed into the learning rate, making its value irrelevant to the optimization outcome.The implementation used the iR-prop gradient-based optimizer.
- Validation: The optimization was performed independently on 12 data chunks, fitting the first 32 s and validating on the remaining 8 s of each chunk.Validation likelihood evaluates inferred parameters on data not used for fitting.
Excitatory - Inhibitory module, the effective transfer function
The model separates excitatory and inhibitory populations while using the excitatory recording as the observable constraint. Faster inhibitory dynamics are adiabatically eliminated, and spatial connectivity is regularized by anatomical priors.
- Excitatory–inhibitory structure: Each pixel contains excitatory and inhibitory sub-populations, but the calcium indicator is assumed to provide a proxy for excitatory activity only.Inhibitory activity is therefore treated as a hidden variable.
- Effective transfer function: The effective excitatory transfer function replaces the unobserved inhibitory activity by evaluating it at its stationary value for fixed excitatory activity.This adiabatic approximation assumes inhibitory neurons operate on a faster time scale.
- Effective transfer function: Under the effective transfer function, kee and kei are optimized while kie and kii are fixed at −25 mV.The approximation removes the dependence on the latter two couplings from the likelihood.
- Connectivity priors: Lateral connectivity is constrained to decay exponentially with distance, reducing the number of parameters in the inferred model.The spatial scale λ parameterizes this decay, with distances between populations entering the connectivity kernel.
- Connectivity priors: The distance between populations is modeled with directional and anisotropic terms involving ρik, ek, ak, θik, and φk.These terms specify the spatial structure used in the connectivity parametrization.
Gaussian Mixture Model
Slow waves are represented by transition-time vectors and clustered with Gaussian mixture models to infer typical propagation modes. The number of modes is selected by comparing fits across candidate cluster counts.
- Gaussian Mixture Model: Each slow wave is encoded as a vector of transition times to the up state across 44 or 41 informative channels.The resulting high-dimensional vectors support classification of spatio-temporal propagation patterns.
- Gaussian Mixture Model: A Gaussian mixture assigns each wave to one of K propagation patterns, each described by a mean vector, covariance matrix, and mixture weight.The total likelihood sums over the possible cluster assignments for all waves.
- Gaussian Mixture Model: Expectation–Maximization estimates mixture parameters by maximizing the likelihood of the observed wave vectors.The procedure uses sklearn.mixture.GaussianMixture to infer modes and assign waves.
- Gaussian Mixture Model: Because EM can converge to different local maxima, reshuffling the data and averaging final cluster configurations can reduce initialization dependence.This issue arises because the clustering problem is non-convex.
- Model selection: The unknown number of propagation patterns is evaluated by running the clustering procedure for different K values and comparing Gaussianity-test performance.The Gaussianity test provides a quantitative comparison across candidate cluster counts.
- Outer-loop modulation: Periodic neuromodulation varies the parameters b and Iext using an amplitude A and period T optimized in the outer loop.The baseline parameter set is supplied by the inner-loop inference.
The Outer Loop: grid search and data-simulation comparison
The outer loop searches neuro-modulation parameters by comparing simulated and experimental distributions of cortical slow-wave observables. A shared analysis pipeline extracts wave speed, direction, and inter-wave interval, while EMD-based scoring and biological constraints identify suitable simulations.
- Grid search: The outer loop evaluates candidate amplitude A and period T values through a grid search spanning 22 amplitude and 21 period settings.A ranges from 0.0 to 6, while T ranges from 0.37 s to 6.9 s.
- Observable extraction: Each simulation is processed with the same pipeline as experimental data to obtain distributions of wave speed, propagation direction, and inter-wave interval.The pipeline includes processing, trigger detection, wave detection, and wave characterization.
- Data-simulation comparison: EMD quantifies the distance between simulated and experimental distributions, and the three observable-specific distances are combined into an Euclidean similarity score.EMD represents the minimum work required to transform one distribution into another and is evaluated using discretely binned cumulative distributions.
- Biological constraints: Biological constraints reject simulations whose down-state or wave-duration distributions exceed experimental minimum-based percentile criteria.The acceptance criteria use the 98th percentile for down-state duration and the 95th percentile for wave duration after outlier removal.
- Wave characterization: Wave characterization measures local velocity, direction, and slow-oscillation frequency from detected transition-time patterns across pixels.Local directions use Gaussian-weighted velocity components, while slow-oscillation frequency is computed from the lag between consecutive wave transitions.
Code availability
The study provides implementation and reproducibility resources, including source code and publicly available experimental recordings.
- Code availability: The inference implementation, data-analysis code, and figure-reproduction scripts are available on GitHub.The repository is described as supporting reproduction of the paper’s figures.
- Data availability: The experimental wide-field calcium-imaging recordings come from publicly available EBRAINS Knowledge Graph datasets of anesthetized mice.These recordings are identified as the experimental data source.
- Declarations: The authors declare no competing interests.
Supporting information
Supporting analyses characterize trial variability, compare direct likelihood inference with the proposed two-step approach, and discuss why neuro-modulation inference is difficult.
- Trial variability: Experimental wave velocities, directions, and inter-wave intervals vary across trials from the same mouse, with differences potentially related to anaesthesia level.The variability is quantified using cumulative distributions and Earth Mover’s Distance.
- Likelihood inference: Direct likelihood inference of neuro-modulation parameters does not fully reproduce the data’s temporal non-stationarity, particularly its inter-wave-interval distribution.The two-step inner-plus-outer-loop approach is contrasted with a single inference step that includes neuro-modulation in the likelihood.
- Model comparison: Inferred neuro-modulation amplitudes and periods are consistent across temporal recording chunks, and model dynamics are directly compared with data for both mice.These comparisons are reported separately for each trial and mouse.
- Limitation: Direct inner-loop inference may require estimating the correct neuro-modulation phase, which is harder when the signal is noisy and recordings are limited.The authors suggest that a larger amount of data would be required.
A posteriori validation of the simulated wave modes
A posteriori validation compares simulated wave modes with experimental modes and a shuffled control. The optimal simulations reproduce the three principal experimental modes, while a less frequent fourth mode is more prominent in simulations.
- Quantitative validation: Simulated datasets achieve confidence values of 88 ± 2% for mouse 1 and 85 ± 1% for mouse 2 in the mode representation analysis.The values are obtained from testing-data log-likelihood comparisons using GMM modes fitted on training experimental waves.
- Validation design: The validation compares optimal simulations with a channel-shuffled control to assess the statistical stability of simulated wave modes.The comparison is performed separately for the two mice.
- Experimental modes: Three propagation modes are identified when the Gaussian mixture model is fitted to experimental waves alone.The number of modes is selected automatically using the Bayesian Information Criterion.
- Data-simulation comparison: When experimental and simulated waves are combined, the three main modes occur in both datasets, while a less frequent fourth mode predominates in simulations.This occupancy pattern is reported for both mice.
- Control comparison: The shuffled simulation produces a different occupancy distribution, with experimental modes represented by only two identified modes and the predominant mode absent from shuffled simulations.
Detecting the number of components in the GMM
The number of GMM components is selected automatically with the Bayes Information Criterion, using likelihood maximization to assess fit quality.
- The GMM component count is automatically selected using the Bayes Information Criterion (BIC) within a likelihood maximization protocol.Lower BIC scores indicate better prediction of the available data and, by extension, the unknown data distribution.
Simulation with pulse stimulation
The simulations show that neuromodulation controls whether focal stimulation produces no propagation, a non-trivial pattern, or a global traveling wave. The outer loop reproduces experimentally observed non-stationary dynamics, while simulated wave modes are validated against experimental and shuffled controls.
- Pulse stimulation: Low excitability prevents propagation, intermediate excitability produces a non-trivial wave pattern, and high excitability generates a global traveling wave after focal stimulation.These conditions correspond to the top, middle, and bottom panels of the stimulation simulations.
- Pulse stimulation: Even with connectivity inferred from a single brain state, the model supports propagation patterns ranging from spirals and traveling waves to asynchronous activity.The repertoire spans deep anesthesia states through the transition toward dissolution of slow-wave patterns.
- Non-stationary dynamics: The outer loop reproduces non-stationarity seen in experimental data, whereas the inner loop alone produces stationary dynamics.Spectrograms and normalized frequency auto-correlation distinguish the inner-loop output from both experimental and outer-loop simulation data.
- A posteriori validation: Figure S3 compares wave modes identified from experimental-only, combined experimental-and-simulated, and experimental-and-shuffled-simulated data across two mice.Columns report mode fractions for combined collections and their experimental, simulated, or shuffled-simulated subsets; channel shuffling uses one common permutation.
- GMM component selection: Figure S4 uses BIC scores and their gradients to identify optimal GMM component counts separately for each mouse and dataset.Blue denotes experimental data, orange denotes experimental-plus-simulated data, and vertical dotted lines mark the optima.
- Non-stationary dynamics: The normalized mean frequency auto-correlation decreases rapidly for experimental and outer-loop data but remains high for inner-loop output.The same pattern is consistent across trials and individuals.