Ten rules for effective modeling.

Modeling a natural system such as the brain aims to deepen understanding and to help to explain and link multiple phenomena into a coherent picture. In any specific case, this requires a clear view of the aims of each modeling project, followed by coordinated selection of the model's style and components; theoretical, numerical, and statistical analysis methods; distillation and presentation of results; and resulting well supported conclusions. The ten rules presented here apply to modeling of the brain and other systems and are designed to assist in carrying out integrated modeling with valid and well-supported outcomes that effectively achieve the modeling aims; referees can also use them when assessing the validity of modeling in submitted manuscripts.

Generalized neural field theory of cortical plasticity illustrated by an application to the linear phase of ocular dominance column formation in primary visual cortex.

Physiologically based neural field theory (NFT) is extended to encompass cortical plasticity dynamics. An illustrative application is provided which treats the evolution of the connectivity of left- and right-eye visual stimuli to neuronal populations in the primary visual cortex (V1), and the initial, linear phase of formation of approximately one-dimensional (1D) ocular dominance columns (ODCs) that sets their transverse spatial scale. This links V1 activity, structure, and physiology within a single theory that already accounts for a range of other brain activity and connectivity phenomena, thereby enabling ODC formation and many other phenomena to be interrelated and cortical parameters to be constrained across multiple domains. The results accord with experimental ODC widths for realistic cortical parameters and are based directly on a unified description of the neuronal populations involved, their connection strengths, and the neuronal activity they support. Other key results include simple analytic approximations for ODC widths and the parameters of maximum growth rate, constraints on cortical excitatory and inhibitory gains, elucidation of the roles of specific poles of the V1 response function, and the fact that ODCs are not formed when input stimuli are fully correlated between eyes. This work provides a basis for further generalization of NFT to model other plasticity phenomena, thereby linking them to the range multiscale phenomena accounted for by NFT.

Neural field theory of neural avalanche exponents.

The power-law exponents of observed size and lifetime distributions of near-critical neural avalanches are calculated from neural field theory using diagrammatic methods. This brings neural avalanches within the ambit of neural field theory, which has also previously explained near-critical 1/f spectra and many other observed features of neural activity. This strengthens the case for near-criticality of the brain and opens the way for these other phenomena to be interrelated with avalanches and their dynamics.

Relationships between lognormal distributions of neural properties, activity, criticality, and connectivity.

Relationships between convergence of inputs onto neurons, divergence of outputs from them, synaptic strengths, nonlinear firing response properties, and randomness of axonal ranges are systematically explored by interrelating means and variances of synaptic strengths, firing rates, and soma voltages. When self-consistency is imposed, it is found that broad distributions of synaptic strength are a necessary concomitant of the known massive convergence of inputs to individual neurons, and observed widths of lognormal distributions of synaptic strength and firing rate are explained provided the brain is in a near-critical state, consistent with independent observations. The strongest individual synapses are shown to have an effect on soma voltage comparable to the effect of all others combined, which supports suggestions that they may have a key role in neural communication. Remarkably, inclusion of moderate randomness in characteristic axonal ranges is shown to account for the observed [Formula: see text]-fold variability in two-point connectivity at a given separation and [Formula: see text]-fold overall when the known mean exponential fall-off is included, consistent with observed near-lognormal distributions. Inferred axonal deviations from straight-line paths are also consistent with independent estimates.

Nonlinear wave-wave interactions in the brain.

Neural field theory of the corticothalamic system is used to analyze nonlinear wave-wave interactions in steady state visual evoked potential responses. The nonlinear power spectrum is analytically calculated by convolving the linear power spectrum with itself and other factors. Periodic sine and square wave stimuli are used to generate steady state visual evoked potential responses and to study stimulus-driven nonlinear corticothalamic dynamic interactions. Moreover, we use dual sine drives to analyze the driven dynamics. Numerical analysis shows that the nonlinear power spectrum embodies key nonlinear features, including harmonic and subharmonic generation, entrainment of the alpha rhythm to periodic stimuli at the drive frequency, sum and difference frequencies due to wave-wave coalescence and decay. Further, the scaling properties of the key phenomena observed in nonlinear interactions are studied, verifying some of the theoretical predictions for these being generated by three-wave processes.

Importance of self-connections for brain connectivity and spectral connectomics.

Spectral analysis and neural field theory are used to investigate the role of local connections in brain connectivity matrices (CMs) that quantify connectivity between pairs of discretized brain regions. This work investigates how the common procedure of omitting such self-connections (i.e., the diagonal elements of CMs) in published studies of brain connectivity affects the properties of functional CMs (fCMs) and the mutually consistent effective CMs (eCMs) that correspond to them. It is shown that retention of self-connections in the fCM calculated from two-point activity covariances is essential for the fCM to be a true covariance matrix, to enable correct inference of the direct total eCMs from the fCM, and to ensure their compatibility with it; the deCM and teCM represent the strengths of direct connections and all connections between points, respectively. When self-connections are retained, inferred eCMs are found to have net inhibitory self-connections that represent the local inhibition needed to balance excitation via white matter fibers at longer ranges. This inference of spatially unresolved connectivity exemplifies the power of spectral connectivity methods, which also enable transformation of CMs to compact diagonal forms that allow accurate approximation of the fCM and total eCM in terms of just a few modes, rather than the full [Formula: see text] CM entries for connections between N brain regions. It is found that omission of fCM self-connections affects both local and long-range connections in eCMs, so they cannot be omitted even when studying the large-scale. Moreover, retention of local connections enables inference of subgrid short-range inhibitory connectivity. The results are verified and illustrated using the NKI-Rockland dataset from the University of Southern California Multimodal Connectivity Database. Deletion of self-connections is common in the field; this does not affect case-control studies but the present results imply that such fCMs must have self-connections restored before eCMs can be inferred from them.

Nonlinear harmonic generation in the corticothalamic system.

Neural field theory of the corticothalamic system is applied to quantitatively analyze harmonic generation in normal sleep and wake states. The linear power spectrum is derived analytically via the transfer function and is then convolved with itself and other factors to calculate the nonlinear power spectrum analytically via a recent perturbation expansion. Analysis shows that strong spectral peaks generate a harmonic at twice the original frequency with peak power proportional to the square of that of the original peak. Fits to the data enable absolute normalization to be determined, with the conclusion that the experimentally observed spindle harmonic peak is nonlinear. Using this normalization, the same analysis is applied to the wake state and nonlinear contributions to the alpha and beta peaks are quantified.

Neural mechanisms of the EEG alpha-BOLD anticorrelation.

An experimentally tested neural field theory of the corticothalamic system is used to model brain activity and resulting experimental EEG data, and to elucidate the neural mechanisms and physiological basis of alpha-BOLD anticorrelation observed in concurrent EEG and fMRI measurements. Several studies have proposed that the anticorrelation originates from a causal link between changes in the alpha power and BOLD signal. However, the results in this study reveal that fluctuations in alpha and BOLD power do not generate one another but instead respectively result from high- and low-frequency components of the same underlying cortical activity, and that they are inversely correlated via variations in the strengths of corticothalamic and intrathalamic feedback, thereby explaining their anticorrelation.

Dependence of absence seizure dynamics on physiological parameter evolution.

A neural field model of the corticothalamic system is applied to investigate the temporal and spectral characteristics of absence seizures in the presence of a temporally varying connection strength between the cerebral cortex and thalamus. Increasing connection strength drives the system into an absence seizure-like state once a threshold is passed and a supercritical Hopf bifurcation occurs. The dynamics and spectral characteristics of the resulting model seizures are explored as functions of maximum connection strength, time above threshold, and the rate at which the connection strength increases (ramp rate). Our results enable spectral and temporal characteristics of seizures to be related to changes in the underlying physiological evolution of connections via nonlinear dynamics and neural field theory. Spectral analysis reveals that the power of the harmonics and the duration of the oscillations increase as the maximum connection strength and the time above threshold increase. It is also found that the time to reach the stable limit-cycle seizure oscillation from the instability threshold decreases with the square root of the ramp rate.

Physiology-based ERPs in normal and abnormal states.

Evoked response potentials (ERPs) and other transients are modeled as impulse responses using physiology-based neural field theory (NFT) of the corticothalamic system of neural activity in the human brain that incorporates synaptic and dendritic dynamics, firing response, axonal propagation, and corticocortical and corticothalamic pathways. The properties of model-predicted ERPs are explored throughout the stability zone of the corticothalamic system, and predicted time series and wavelet spectra are also analyzed. This provides a unified treatment of predicted ERPs for both normal and abnormal states within the brain's stability zone, including likely parameters to represent abnormal states of reduced arousal.

Effects of astrocytic dynamics on spatiotemporal hemodynamics: Modeling and enhanced data analysis.

The effects of astrocytic dynamics on the blood oxygen-level dependent (BOLD) response are modeled. The dynamics are represented via an astrocytic response function that approximates the effects of astrocytic activity, including delay between neural activity and hemodynamic response. The astrocytic response function is incorporated into a spatiotemporal hemodynamic model to predict the BOLD response measured using functional magnetic resonance imaging (fMRI). Adding astrocytic dynamics is shown to significantly improve the ability of the model to robustly reproduce the spatiotemporal properties of the experimental data such as characteristic frequency and time-to-peak. Moreover, the results are consistent across different astrocytic response functions, thus a simple impulsive form suffices to model the effective time delay of astrocytic responses. Finally, the results yield improved estimates of previously reported hemodynamic parameters, such as natural frequency and decay rate of the flow signal, which are consistent with experimentally verified physiological limits. The techniques developed in this study will contribute to improved analysis of BOLD-fMRI data.

Neural field model of seizure-like activity in isolated cortex.

Epileptiform discharges on an isolated cortex are explored using neural field theory. A neural field model of the isolated cortex is used that consists of three neural populations, excitatory, inhibitory, and excitatory bursting. Mechanisms by which an isolated cortex gives rise to seizure-like waveforms thought to underly pathological EEG waveforms on the deafferented cortex are explored. It is shown that the model reproduces similar time series and oscillatory frequencies for paroxysmal discharges when compared with physiological recordings both during acute and chronic deafferentation states. Furthermore, within our model ictal activity arises from perturbations to steady-states very close to the dynamical system's instability boundary; hence, these are distinct from corticothalamic seizures observed in the model for the intact brain which involved limit-cycle dynamics. The results are applied to experiments in deafferented cats.

Multistability in the corticothalamic system.

Neural field theory of the corticothalamic system is used to analyze the properties of its steady-state solutions, including their linear stability, in the parameter space of synaptic couplings for physiological parameter ranges representing normal arousal waking states in adult humans. The independent connections of the corticothalamic model define an eight-dimensional parameter space, while specific combinations of these connections parameterize intracortical, corticothalamic, and intrathalamic loops. Multistable regions are systematically identified and the existence of up to five steady-state solutions is confirmed, up to three of which are linearly stable. A key determinant for the existence of five steady states is found to be the number of nonzero connections. This finding had not been previously proposed as the determining factor of high multiplicities of multistability in mesoscopic models of the brain. In the corticothalamic model presented here, multistability occurs when the intrathalamic loop is present (i.e., the reticular nucleus inhibits the relay nuclei), and when the net synaptic effect of the intracortical loop is inhibitory. The signature of these additional waking states is an overall increased level of thalamic activity. It is argued that the additional steady states found may represent hyperarousal states which occur when the corticothalamic projections do not attenuate the activity of the cortex.

Unified neural field theory of brain dynamics underlying oscillations in Parkinson's disease and generalized epilepsies.

The mechanisms underlying pathologically synchronized neural oscillations in Parkinson's disease (PD) and generalized epilepsies are explored in parallel via a physiologically-based neural field model of the corticothalamic-basal ganglia (CTBG) system. The basal ganglia (BG) are approximated as a single effective population and their roles in the modulation of oscillatory dynamics of the corticothalamic (CT) system and vice versa are analyzed. In addition to normal EEG rhythms, enhanced activity around 4 Hz and 20 Hz exists in the model, consistent with the characteristic frequencies observed in PD. These rhythms result from resonances in loops formed between the BG and CT populations, analogous to those that underlie epileptic oscillations in a previous CT model, and which are still present in the combined CTBG system. Dopamine depletion is argued to weaken the dampening of these loop resonances in PD, and network connections then explain the significant coherence observed between BG, thalamic, and cortical population activity around 4-8 Hz and 20 Hz. Parallels between the afferent and efferent connection sites of the thalamic reticular nucleus (TRN) and BG predict low dopamine to correspond to a reduced likelihood of tonic-clonic (grand mal) seizures, which agrees with experimental findings. Furthermore, the model predicts an increased likelihood of absence (petit mal) seizure resulting from pathologically low dopamine levels in accordance with experimental observations. Suppression of absence seizure activity is demonstrated when afferent and efferent BG connections to the CT system are strengthened, which is consistent with other CTBG modeling studies. The BG are demonstrated to have a suppressive effect on activity of the CTBG system near tonic-clonic seizure states, which provides insight into the reported efficacy of current treatments in BG circuits. Sleep states of the TRN are also found to suppress pathological PD activity in accordance with observations. Overall, the findings demonstrate strong parallels between coherent oscillations in generalized epilepsies and PD, and provide insights into possible comorbidities.

K-complexes, spindles, and ERPs as impulse responses: unification via neural field theory.

To interrelate K-complexes, spindles, evoked response potentials (ERPs), and spontaneous electroencephalography (EEG) using neural field theory (NFT), physiology-based NFT of the corticothalamic system is used to model cortical excitatory and inhibitory populations and thalamic relay and reticular nuclei. The impulse response function of the model is used to predict the responses to impulses, which are compared with transient waveforms in sleep studies. Fits to empirical data then allow underlying brain physiology to be inferred and compared with other waves. Spontaneous K-complexes, spindles, and other transient waveforms can be reproduced using NFT by treating them as evoked responses to impulsive stimuli with brain parameters appropriate to spontaneous EEG in sleep stage 2. Using this approach, spontaneous K-complexes and sleep spindles can be analyzed using the same single theory as previously been used to account for waking ERPs and other EEG phenomena. As a result, NFT can explain a wide variety of transient waveforms that have only been phenomenologically classified to date. This enables noninvasive fitting to be used to infer underlying physiological parameters. This physiology-based model reproduces the time series of different transient EEG waveforms; it has previously reproduced experimental EEG spectra, and waking ERPs, and many other observations, thereby unifying transient sleep waveforms with these phenomena.

The spatiotemporal hemodynamic response function for depth-dependent functional imaging of human cortex.

The gray matter of human cortex is characterized by depth-dependent differences in neuronal activity and connections (Shipp, 2007) as well as in the associated vasculature (Duvernoy et al., 1981). The resolution limit of functional magnetic resonance imaging (fMRI) measurements is now below a millimeter, promising the non-invasive measurement of these properties in awake and behaving humans (Muckli et al., 2015; Olman et al., 2012; Ress et al., 2007). To advance this endeavor, we present a detailed spatiotemporal hemodynamic response function (HRF) reconstructed through the use of high-resolution, submillimeter fMRI. We decomposed the HRF into directions tangential and perpendicular to the cortical surface and found that key spatial properties of the HRF change significantly with depth from the cortical surface. Notably, we found that the spatial spread of the HRF increases linearly from 4.8mm at the gray/white matter boundary to 6.6mm near the cortical surface. Using a hemodynamic model, we posit that this effect can be explained by the depth profile of the cortical vasculature, and as such, must be taken into account to properly estimate the underlying neuronal responses at different cortical depths.

Eigenmodes of brain activity: Neural field theory predictions and comparison with experiment.

Neural field theory of the corticothalamic system is applied to predict and analyze the activity eigenmodes of the bihemispheric brain, focusing particularly on their spatial structure. The eigenmodes of a single brain hemisphere are found to be close analogs of spherical harmonics, which are the natural modes of the sphere. Instead of multiple eigenvalues being equal, as in the spherical case, cortical folding splits them to have distinct values. Inclusion of interhemispheric connections between homologous regions via the corpus callosum leads to further splitting that depends on symmetry or antisymmetry of activity between brain hemispheres, and the strength and sign of the interhemispheric connections. Symmetry properties of the lowest observed eigenmodes strongly constrain the interhemispheric connectivity strengths and unihemispheric mode spectra, and it is predicted that most spontaneous brain activity will be symmetric between hemispheres, consistent with observations. Comparison with the eigenmodes of an experimental anatomical connectivity matrix confirms these results, permits the relative strengths of intrahemispheric and interhemispheric connectivities to be approximately inferred from their eigenvalues, and lays the foundation for further experimental tests. The results are consistent with brain activity being in corticothalamic eigenmodes, rather than discrete "networks" and open the way to new approaches to brain analysis.

Calcium dependent plasticity applied to repetitive transcranial magnetic stimulation with a neural field model.

The calcium dependent plasticity (CaDP) approach to the modeling of synaptic weight change is applied using a neural field approach to realistic repetitive transcranial magnetic stimulation (rTMS) protocols. A spatially-symmetric nonlinear neural field model consisting of populations of excitatory and inhibitory neurons is used. The plasticity between excitatory cell populations is then evaluated using a CaDP approach that incorporates metaplasticity. The direction and size of the plasticity (potentiation or depression) depends on both the amplitude of stimulation and duration of the protocol. The breaks in the inhibitory theta-burst stimulation protocol are crucial to ensuring that the stimulation bursts are potentiating in nature. Tuning the parameters of a spike-timing dependent plasticity (STDP) window with a Monte Carlo approach to maximize agreement between STDP predictions and the CaDP results reproduces a realistically-shaped window with two regions of depression in agreement with the existing literature. Developing understanding of how TMS interacts with cells at a network level may be important for future investigation.

Generalized seizures in a neural field model with bursting dynamics.

The mechanisms underlying generalized seizures are explored with neural field theory. A corticothalamic neural field model that has accounted for multiple brain activity phenomena and states is used to explore changes leading to pathological seizure states. It is found that absence seizures arise from instabilities in the system and replicate experimental studies in numerous animal models and clinical studies.

Slow-wave oscillations in a corticothalamic model of sleep and wake.

A physiologically-based corticothalamic neural field model is used to study slow wave oscillations including cortical UP and DOWN states in deep sleep by extending it to incorporate bursting dynamics of neurons in the thalamic reticular nucleus. The interplay of local bursting dynamics and network interactions produces the cortical UP and DOWN states of slow wave sleep while preserving previously verified model predictions in the wake state. Results show that EEG spectral features in wake and sleep are reproduced. The bursting is subthreshold but acts to intensify the amplitude of oscillations in slow wave sleep with deep UP/DOWN oscillations on the cortex emerging naturally. Furthermore, there is a continuous cycle between the two regimes, rather than a flip-flop between discrete states.