# Wilson–Cowan Equations for Neocortical Dynamics

- Jack D. Cowan
^{1}Email author, - Jeremy Neuman
^{2}and - Wim van Drongelen
^{3}

**6**:1

https://doi.org/10.1186/s13408-015-0034-5

© Cowan et al. 2016

**Received: **27 July 2015

**Accepted: **18 December 2015

**Published: **4 January 2016

## Abstract

In 1972–1973 Wilson and Cowan introduced a mathematical model of the population dynamics of synaptically coupled excitatory and inhibitory neurons in the neocortex. The model dealt only with the mean numbers of activated and quiescent excitatory and inhibitory neurons, and said nothing about fluctuations and correlations of such activity. However, in 1997 Ohira and Cowan, and then in 2007–2009 Buice and Cowan introduced Markov models of such activity that included fluctuation and correlation effects. Here we show how both models can be used to provide a quantitative account of the population dynamics of neocortical activity.

We first describe how the Markov models account for many recent measurements of the resting or spontaneous activity of the neocortex. In particular we show that the power spectrum of large-scale neocortical activity has a Brownian motion baseline, and that the statistical structure of the random bursts of spiking activity found near the resting state indicates that such a state can be represented as a percolation process on a random graph, called *directed* percolation.

Other data indicate that resting cortex exhibits pair correlations between neighboring populations of cells, the amplitudes of which decay slowly with distance, whereas stimulated cortex exhibits pair correlations which decay rapidly with distance. Here we show how the Markov model can account for the behavior of the pair correlations.

Finally we show how the 1972–1973 Wilson–Cowan equations can account for recent data which indicates that there are at least two distinct modes of cortical responses to stimuli. In mode 1 a low intensity stimulus triggers a wave that propagates at a velocity of about 0.3 m/s, with an amplitude that decays exponentially. In mode 2 a high intensity stimulus triggers a larger response that remains local and does not propagate to neighboring regions.

### Keywords

Wilson–Cowan equations Bogdanov–Takens bifurcation Propagating decaying LFP and VSD waves Localized decaying LFP and VSD responses Neural network master equation Directed percolation phase transition Pair-correlations## 1 Introduction

The analysis of large-scale brain activity is a difficult problem. There are about 50 billion neurons in the cortex of the human brain: 80 % are excitatory, whereas the remaining 20 % are inhibitory. Each neuron has about seven thousand axon terminals from other neurons, but there is some redundancy in the connectivity so that it has effective connections from about 80 other neurons, mostly nearest neighbors. Each neuron is actually a complex switching device, but in this review, we introduce only the simplest cellular model, that neurons are binary switches, either quiescent or activated. It follows that there are approximately \(10^{1.5 \times10^{10}}\) configurations of activated or quiescent neurons. Such a large configuration space suggests the need to use statistical methods to analyze large-scale brain activity. In addition there is some degree of microscopic randomness in neural connectivity, and there are also random fluctuations of neural activity, both of which also support the need for a statistical treatment, as noted by Sholl in 1956 [1].

## 2 Experimental Data on Large-Scale Brain Activity

There is a large body of data on large-scale brain activity, including electroencephalographic (EEG) recordings with large electrodes from the surface of the scalp, functional magnetic resonance (fMRI) measurements of blood flow in different brain regions (also large-scale), local field potentials (LFP) recorded with smaller electrodes, microelectrode recordings from or near individual neurons, or (currently) microelectrode arrays which can record the simultaneous activity of many neighboring neurons. Currently there are also new techniques for forming optical images of local brain activity, using voltage sensitive dyes (VSD). All such recordings can be classified as either *spontaneous* or *resting* activity, or stimulus-driven *evoked* activity.

### 2.1 Resting Activity

*alpha*rhythm, seen in awake relaxed humans, mainly in the occipital region of the brain which processes visual signals from the eyes. Figure 2 shows the power spectrum of such activity. It will be seen that there is a pronounced peak in the power spectrum at around 10 Hz and a secondary peak around 20 Hz. This peak is said to be in the range of the

*beta*rhythm of occipital EEG activity. Interestingly if the contributions of such peaks are eliminated, what is left can be fitted with the function \(a/(b+f^{2})\), where

*a*and

*b*are constants, and

*f*is the frequency in Hz. Figure 3 shows such a function and its fit to the EEG power spectrum. It is important to note that this power spectrum fit is that of Brownian motion, which suggests that resting brain activity is largely

*desynchronized*and

*random*.

#### 2.1.1 Isolated Neocortex

But the most detailed studies, and the most information about the nature of spontaneous activity, has been obtained from studies of isolated neocortical slabs. The first detailed studies were carried out in the early 1950s by DeLisle Burns, on isolated slabs of parietal neocortex [7, 8]. The main relevant result was that very lightly anesthetized slabs spontaneously generated *bursts* of propagating activity from a number of randomly occurring sites. Any variation of the level of anesthesia, either up or down, abolished the activity.

Beggs and Plenz’s conclusion is that such bursts of activity are *avalanches* defined as follows: the configuration of active electrodes in the array during one time bin of width Δ*t* is termed a frame, and a sequence of frames preceded and followed by blank frames is called an avalanche. However, successive frames are not highly correlated, so the activity is not wave-like: it is in fact *self-similar*, and in addition, the avalanche size distribution follows the power law \(P[n] \propto n^{\alpha}\). In addition the exponent *α* is approximately −1.5. This is the mean-field exponent of a critical branching process [10]. This result was a step beyond that of Softky and Koch [11] who found Poisson-like spiking activity in individual cortical neurons, and introduced the possibility of criticality in brain dynamics. In fact this mean-field exponent turns up in several kinds of *percolation* processes on random graphs, including both isotropic and directed percolation. But branching and annihilating random walks are equivalent to directed percolation, so it is possible that what Beggs and Plenz observed in cortical slices was a form of directed percolation. We will return to this topic later.

### 2.2 Driven or Stimulated Activity

In case there is an external stimulus, neocortical dynamics indicates a very different picture. It turns out that there is a big difference in the responses to weak stimuli, compared to those triggered by stronger stimuli. In addition correlations between pairs of neurons in driven neocortex have a shorter length scale than those found in spontaneous activity.

#### 2.2.1 Weak Stimuli

*propagating wave*whose amplitude decays exponentially with distance. Figure 7 shows the cortical responses to low amplitude stimuli in the form of spikes, recorded by an implanted microelectrode array in three monkey visual cortices by Nauhaus et al. [13]. Each row shows data from the spike-triggered local field potentials (LFP) from a single location. The first column shows the dependence of time to peak of the LFP as a function of the cortical distance from the triggering electrode, and estimated propagation velocities. The second column shows the propagating wave, both as a pseudo-colored image, and as a plot of wave amplitude vs. distance from the triggering electrode, together with estimates of the space-constants of the decaying waves. The third column shows average LFP waveforms at three locations from the triggering spike.

It will be seen that the response is indeed a traveling LFP, whose velocity is about 25–30 cm/s. In addition the LFP amplitude decays exponentially, with a decay constant *λ* of about 3 mm.

#### 2.2.2 Strong Stimuli

*localized*. Figure 8 shows a comparison of cortical responses to weak and strong stimuli [13]. It will be seen that responses to larger stimuli remain essentially

*localized*. These observations immediately suggest a role for inhibition in localizing such responses.

#### 2.2.3 Correlations

To explain all these observations we need to understand the competing roles of neural excitation and inhibition in neural population dynamics. We therefore give a short account of the history and development of the Wilson–Cowan neural population equations.

## 3 Neural Population Equations

### 3.1 Introduction

Following early work by Shimbel and Rapaport [20], Beurle [21] focused, not on the activity of single neurons, but on the proportion of neurons activated per unit time in a given volume element of a slice or slab of neocortex, denoted by \(n(\mathbf {x},t)\). For all practical purposes this can be taken to be equivalent to the spike-triggered LFP and VSD described earlier.

*exactly*threshold excitation. [There is an implicit assumption that individual neurons are of the integrate-and-fire variety.]

- 1.
By assuming that \(n( t+\tau) = q(t)f[n(t)]\) Beurle ignored the effects of fluctuations and correlations on the dynamics. It is not true that

*q*and \(f[n]\) are statistically independent quantities, as was first pointed out in [22]. - 2.
The update equation is incorrect. \(f[n]\) should be the proportion of neurons receiving

*at least*threshold excitation, as was first noted by Uttley [23].This proportion can be expressed [24] as:where \(\vartheta[n]\) is the Heaviside step function and \(\langle \vartheta[n] \rangle\) is the average of \(\vartheta[n]\) over the probability distribution of thresholds \(P(n_{\mathrm{TH}})\).$$ f[n] = \int_{-\infty}^{n} P(n_{\mathrm{TH}}) \,dn_{\mathrm{TH}}= \int _{-\infty}^{\infty} \vartheta[n-n_{\mathrm{TH}}]P(n_{\mathrm{TH}}) \,dn_{\mathrm{TH}}=\bigl\langle \vartheta[n] \bigr\rangle , $$(2)This implies that the function \(f[n]\) should have the form of a probability distribution function, not a probability density. In Cowan [25] the logistic or*sigmoid*form,was introduced, as an analytic approximation to the Heaviside step function used in McCulloch–Pitts neurons [26]. This indicates that the required continuum equations should represent the dynamics of a population of integrate-and-fire neurons in which there is a random distribution of thresholds.$$ f[n] = \bigl[1+ \exp[-n]\bigr]^{-1}=\frac{1}{2} \biggl[1+ \tanh \biggl(\frac{n}{2}\biggr)\biggr] $$(3)The corrected version of Beurle’s equation takes the formwhere$$\begin{aligned} &n(\mathbf{x}, t+\tau) \\ &\quad= q(\mathbf{x}, t)f\bigl[n(\mathbf{x}, t)\bigr] \\ & \quad= q(\mathbf{x}, t) f \biggl[ \int_{-\infty}^{t} dt^{\prime} \int_{-\infty }^{\infty}d\mathbf{x}^{\prime}\alpha \bigl(t-t^{\prime}\bigr)\bigl[\beta\bigl(\mathbf{x}-\mathbf {x}^{\prime}\bigr)n\bigl(\mathbf{x}^{\prime}, t^{\prime}\bigr)+h\bigl(\mathbf{x},t^{\prime}\bigr)\bigr] \biggr] , \end{aligned}$$(4)\(r = 1\mbox{ ms}\) is the (absolute) refractory period or width of the action potential, and$$ q(\mathbf{x}, t)=1- \int_{t-r}^{t} n(\mathbf{x},t); $$(5)are the impulse response function and spatially homogeneous weighting function of the continuum model, with membrane time constant \(\tau\sim 10~\mbox{ms}\), and space constant \(\sigma\sim100~\upmu \mbox{m}\).$$ \alpha\bigl(t-t^{\prime}\bigr)=\alpha_{0} e^{-(t-t^{\prime})/\tau},\quad\quad \beta\bigl(\mathbf {x}-\mathbf{x}^{\prime}\bigr) = b e^{-|\mathbf{x}-\mathbf{x}^{\prime}|/\sigma} $$(6) - 3.
Beurle’s formulation does not explicitly incorporate a role for inhibitory neurons.

### 3.2 The Wilson–Cowan Equations

Note that \(f_{E}[n]\) and \(f_{I}[n]\) are modified versions of the firing rate function \(f[n]\) introduced in Eq. (3), such that \(f_{E}[0]=f_{I}[0]=0\).

### 3.3 Attractor Dynamics

*focus*. In fact by varying the synaptic weights \(w_{EH}\) and \(w_{IH}\) or \(a=w_{EE} w_{II}\) and \(b=w_{IE}w_{EI}\) we can move from one portrait to another. It turns out that there is a substantial literature dealing with the way in which such changes occur, The mathematical technique for analyzing these transformations is bifurcation theory, and it was first applied to neural problems 53 years ago by Fitzhugh [28], but first applied systematically by Ermentrout and Cowan [29–31] in a series of papers on the dynamics of the mean-field Wilson–Cowan equations. Subsequent studies by Borisyuk and Kirillov [32] and Hoppenstaedt and Izhikevich [33] have greatly extended this analysis.

**a**and

**b**, and is therefore of

*codimension*2. In such a bifurcation an equilibrium point can simultaneously become a marginally stable saddle and an Andronov–Hopf point. So at the bifurcation point the eigenvalues of its stability matrix have zero real parts. In addition the right panel of Fig. 11 shows how the fast E-nullcline and the slow I-nullcline intersect. The first point of contact of the two nullclines is the Bogdanov–Takens bifurcation point. The two nullclines remain close together over a large part of the subsequent

*E*–

*I*phase space before diverging. As we will later discuss, this property of the nullclines is closely connected with the existence of a

*balance*between excitatory and inhibitory currents in the network described by the Wilson–Cowan equations, and therefore with the existence of

*avalanches*in stochastic Wilson–Cowan equations [35].

## 4 Stochastic Neural Dynamics

### 4.1 Introduction

*N*excitatory binary neurons. Such neurons transition from a quiescent state

*q*to an activated state

*a*at the rate

*f*and back again to the quiescent state

*q*at the rate

*α*, as shown in Fig. 12.

### 4.2 A Master Equation for a Network of Excitatory Neurons

*n*activated neurons, each becoming quiescent at the rate

*α*. This produces a flow out of the state

*n*at rate

*α*, proportional to \(p_{n}(t)\), hence a term in the master equation of the form \(-\alpha nP_{n}(t)\). Similarly the flow into

*n*from the state \(n+1\) produces a term \(\alpha(n+1) P_{n+1}(t)\). The net effect is the term

*n*, each prepared to spike at rate \(f[s_{E} (n)]\), leading to the term \(-(N-n)f[s_{E} (n)] P_{n} (t)\), in which the total input is \(s_{E} (n)=I(n)/I_{\mathrm{TH}} = (w_{EE} n + h_{E})/I_{\mathrm{TH}}\), and \(f[s_{E} (n)]\) is the function shown in Fig. 13, a low-noise version of Eq. (3).

*n*from the state \(n-1\) is therefore \((N-n+1)\times f[s_{E} (n-1)] P_{n-1} (t)\), and the total contribution from excitatory spikes is then

*density*of active neurons at the cortical location

**x**at time

*t*, and the total input current \(I(n)\) becomes the current density

### 4.3 A Master Equation for a Network of Excitatory and Inhibitory Neurons

## 5 Analyzing Intrinsic Fluctuations

To analyze such effects we need to look more closely at the attractor dynamics of Eq. (7). There are two cases to consider. In case 1, the attractor is either an asymptotically stable node or focus, or else a limit cycle. In case 2, the attractor is only marginally stable. In nonlinear dynamics this is a bifurcation point, e.g. a Bogdanov–Takens point, or a saddle node or Andronov–Hopf point. In statistical mechanics this is the critical point of a phase transition.

### 5.1 The System-Size Expansion

*N*, the total number of neurons in the network, and standard distribution proportional to \(\sqrt {N}\). So the number of neurons activated at a given time can be represented by the variable

*A*is a constant and \(\eta_{E}\) is an independent white noise variable, whose amplitudes are calculated from Eq. (7).

*dt*are random variables dependent upon the network state. The simulation is carried out for a network in which certain symmetry conditions are introduced. These conditions are

### 5.2 Symmetries and Power Laws

It will be seen that the simulations described above, in which the network symmetry represented in Eq. (17) is present, have uncovered an important property, namely that a stochastic version of Eq. (7) incorporating such a symmetry can spontaneously generate random activity in the form of bursts, whose statistical distribution is a power law. The other important property concerns the basic network dynamics generating such bursts.

We first note the experimental data provided by DeLisle Burns [7] and Beggs and Plenz [9] described in the introduction, and then we discuss the underlying neurodynamics. The main result of the Beggs–Plenz observations is that isolated slices generate bursting behavior similar to that found in the simulations, with a power law burst distribution with slope exponent of \(\beta= -1.5\). This should be compared with the simulation data shown in Fig. 18 in which \(\beta=-1.62\). Note, however, that the geometry of our network simulation is not comparable with that of a cortical slice. It remains to carry out simulations of the stochastic version of Eq. (7) on a 2-dimensional lattice. Work on this is currently ongoing. In any event, the Beggs–Plenz paper generated a great deal of interest in the possibility of critical behavior in the sense of statistical physics existing in stochastic neural dynamics, including the possibility that brain dynamics exhibits self-organized criticality. In the later parts of this paper, we briefly address this possibility.

#### 5.2.1 Random Bursting

*decoupled*, with the unique stable solution \((\varSigma_{0}, 0)\), which is equivalent to \(n_{E} = n_{I}\) in the original variables. This is precisely the stable fixed point used in the simulations. Note also that, in the new variables

*Σ*and Δ, the fixed point current is

*Σ*, and \(\varSigma_{0}\) is unchanged when varying \(w_{E} + w_{I}\) for constant \(w_{0}\). Murphy and Miller called Eq. (20) an

*effective feed-forward system*exhibiting a balance between excitatory and inhibitory currents, and a

*balanced amplification*of a stimulus

*h*.

*A*lies close to the matrix

*B̄*is the signature of the

*normal form*of the Bogdanov–Takens bifurcation [33]. Thus the weakly stable node lies close to a Bogdanov–Takens bifurcation, as we have suggested.

### 5.3 Intrinsic Fluctuations at a Marginally Stable Fixed Point

We now turn to case 2, in which the network dynamics is at a marginally stable fixed point. As we showed earlier, such a fixed point is a Bogdanov–Takens point. We cannot use the system-size expansion at such a point, but we can use the methodology and formalism of statistical field theory [42–45]. However, for the neuro-dynamics considered in this article, case 1 applies: the resting and driven activities are all at or near a weakly stable fixed point. Despite this, the fact that the fixed point is only weakly stable indicates that the resting and weakly driven states lie in what has been called the *fluctuation-driven* region near the marginally stable fixed point [46]. Thus we need to outline some of the results of the analysis of case 2. The reader is referred to the details in the article by Cowan et al. [45].

The basic result is that the stochastic equivalent of the Bogdanov–Takens bifurcation is the critical point of a *Directed Percolation* phase transition, or DP [47]. In DP there are two stable states, separated by a marginally stable critical point. One of these is an *absorbing* state, corresponding to the neural population state in which all neurons are quiescent, so that the mean number of activated states or *order parameter*
\(\langle n \rangle = 0\). The other is one in which many neurons are activated, so that \(\langle n \rangle\neq0\) in the *activated* state. At a critical point the quiescent state becomes marginally stable and is driven by fluctuations into the activated state.

What is important for the present study is that in the neighborhood of such a critical point, i.e. in the fluctuation-driven regime, there are two significant features of the activity which relate to the experimental data we have described: (a) the resting behavior shows random burst behavior whose statistical signature is consistent with DP, i.e., the distribution of bursts follows a power law with slope exponent −1.5, which is the slope of several forms of random percolation, including what is called mean-field DP [9, 10]; (b) intrinsic *correlations* are large, and pair correlations extend over significant cortical distances [18].

## 6 Modeling the Experimental Data

### 6.1 Resting Activity

#### 6.1.1 Random Burst Activity

Assuming that the resting state occurs in the neighborhood of a weakly stable node or focus, to start with we can use the results of the system-size expansion of the *E*–*I* master equation described earlier. The conclusion we reach is that in the case that there is a *balance* between excitation and inhibition, so that the network is at weakly stable node, or possible a focus, then random burst behavior with a power law slope exponent close to −1.5 is seen [35]. This is the result shown in Figs. 14–19, and of course the result is also completely consistent with the Beggs–Plenz data plotted in Figs. 5 and 6. We also note that these results are completely consistent with our recent analysis, Cowan [45], and with recent experimental data that demonstrates the sub-criticality of the resting state by Priesemann et al. [48].

#### 6.1.2 Pair Correlations

As to pair correlations associated with resting or spontaneous activity, we refer to Fig. 9 in which the measured resting pair correlation falls off with pair separation, in both cats and monkeys. This finding can be replicated within the theoretical framework we have established in two differing ways.

*δ*-correlated Gaussian noise to the equations. The resulting pair-correlation function for resting activity is shown in the left panel of Fig. 20. (b) We then use the stochastic Wilson–Cowan master equation introduced in Eq. (14), extended to the spatial case. In such a case the noise is multiplicative and intrinsic, and we used the Gillespie algorithm [38] to simulate the process.

Such simulations of the behavior of Wilson–Cowan equations replicate very accurately, the pair-correlation behavior shown in Fig. 9, reported in [13], both for resting activity and for driven activity.

### 6.2 Driven Activity

#### 6.2.1 Weak Stimuli

#### 6.2.2 Strong Stimuli

The other result reported by Carandini et al. is that for strong stimuli the resulting LFP does *not* propagate very far and remains localized. This property was actually reported in Wilson and Cowan’s 1973 paper [27]! The bottom row of Fig. 21 shows a current simulation of this property, again in which the network state is approximately balanced.

### 6.3 Explaining the Differing Effects of Weak and Strong Stimuli

It is evident that there are big differences between the effects produced by weak and strong stimuli. What is the cause of such differences? Given that the only parameter in the Wilson–Cowan equations that is varied in the two cases is the stimulus intensity, this suggests that the property which causes the different responses is the level of inhibition. It must therefore be the case that the threshold for inhibitory activity is set high enough that weak stimuli do not trigger inhibitory effects, whereas strong enough stimuli do trigger such effects. Indeed this is one of the possibilities suggested by Carandini et al. in their papers. Thus inhibition blocks LFP (and VSD) propagation.

This possibility is also consistent with the effects of stimuli on pair correlations. We predict that the pair-correlation function should falloff more slowly in the case of resting or weakly driven activity, than in the case of stronger stimuli. Such a result would be consistent with the suggestions of Churchland et al. that one effect of stimuli is to lower noise levels.

## 7 Discussion

### 7.1 Early Work

The main results described in this article concern the use of the Wilson–Cowan equations to analyze the dynamics of large populations of interconnected neurons. Early workers, including Shimbel and Rapaport [20] and Beurle [21], appreciated the need to use a statistical formulation of such dynamics, but lacked the techniques to go beyond mean-field theory. The Wilson–Cowan equations [24, 27] were the first major attempt at a statistical theory, but still lacked a treatment of second and higher moments. However, what the equations did describe was mathematical conditions for *attractor* dynamics. Further work by Ermentrout and Cowan [29–31] and by Borisyuk and Kirillov [32], and Hoppenstaedt and Izhikevich [33, 34] used the mathematical techniques of bifurcation theory to more fully analysis such dynamics. The main result was that neural population dynamics is organized around a Bogdanov–Takens bifurcation point, in the neighborhood of which (in a phase space of two control parameters) are saddle-node and Andronov–Hopf bifurcations. Thus neural network dynamics contains locally stable equilibria in the form of stationary and oscillatory attractors.

### 7.2 The System-Size Expansion

The problem of going beyond the mean-field regime proved to be very difficult. Some progress was made by Ohira and Cowan [37] formulating stochastic neural dynamics in the neighborhood of a stable stationary equilibrium as a random Markov process and using the Van Kampen system-size expansion [36]. Further process along these lines was made by Benayoun et al. [35] who formulated Eq. (7) as a random Markov process. But Benayoun et al. went further, by incorporating some symmetries into Eq. (7) discovered by Murphy and Miller [39] which, in retrospect, located the stationary equilibrium of the equations near a Bogdanov–Takens point. The result was that the stochastic version of Eq. (7) generates the random bursts of activity we now refer to as *avalanches*. In addition the avalanche distribution was that of a power law, with a slope exponent \(\beta= 1.6\). This value is close to that observed by Beggs and Plenz [9] in their observations of neural activity in an isolated cortical slab, of avalanche distributions with a slope exponent of \(\beta= 1.5\).

### 7.3 A Statistical Theory of Neural Fluctuations

There remained the problem of developing a statistical theory for the fluctuations about a marginally stable critical point, such as a Bogdanov–Takens point. This problem was formulated by Cowan [42] and solved by Buice and Cowan [43, 44]. This is a major result since it connects the theory of stochastic neural populations at a critical point, with many well studied examples of other populations of interconnected units. Examples include percolation in random graphs, branching and annihilating random walks, catalytic reactions, interacting particles, contact processes, nuclear physics, and bacterial colonies. Many of these processes are subject to a phase transition, known as a directed percolation phase transition (DP). and all these processes have the same statistical properties, including the appearance of random bursts or avalanches.

### 7.4 Relation to Experimental Data

However, although the statistical theory is relevant to the pair-correlation problem, it is the mean-field Wilson–Cowan equations that proved to be necessary and sufficient to analyze neocortical responses to brief stimuli, both weak and strong. In our opinion the close fit between the data and the simulations of the Wilson–Cowan equations with fixed parameters is quite remarkable, especially given the fact that these equations were formulated some 45 to 50 years ago! More detailed papers dealing with these and other results on neocortical responses to stimuli are in preparation.

## Declarations

### Acknowledgements

We thank Dr. Mark Hereld, Argonne National Laboratory, for many helpful discussions and comments. JN was supported in part by the Dr. Ralph and Marian Falk Medical Research Trust and R01 NS084142-01 to Prof. van Drongelen.

**Open Access** This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

## Authors’ Affiliations

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