A canonical microcircuit for estimating Excitation/Inhibition (E/I) balance.

A canonical microcircuit for estimating Excitation/Inhibition (E/I) balance

Excitation/inhibition (E/I) balance is crucial for maintaining healthy brain function and can be disrupted in various neurological and psychiatric disorders. Despite its importance, there are few tools to study E/I balance non-invasively in humans. Here, we propose a canonical microcircuit model to estimate E/I balance from non-invasive magnetoencephalography (MEG), electroencephalography (EEG) or optically pumped magnetometers (OPM) recordings by parameterising global pyramidal and inhibitory cell excitability. We first establish that E/I parameters are identifiable and recoverable. We then explore the effects of these new parameters and their interaction with other parameters in a series of simulations. To highlight the clinical relevance of this new model, we simulate changes in E/I balance and their impact on event-related potentials (ERPs) derived from paired-click, passive and active oddball paradigms, which are among the most robust clinical biomarkers of schizophrenia. Our simulations show that a loss of pyramidal cell excitability can explain reduced ERP amplitudes across all three paradigms, mirroring empirical findings in schizophrenia. This method may serve as a computational assay for estimating synaptopathy and E/I balance from non-invasive electrophysiological recordings across various clinical conditions thereby advancing efforts to develop personalised interventions to restore E/I balance.

Introduction

Our brain relies on a fine-tuned balance between excitation (E) and inhibition (I), referred to as E/I balance, to maintain healthy function. Even small increases in E/I can lead to runaway excitation, as observed in epileptic seizures [1]. Conversely, excessively increased inhibition can lead to sedation - for example, after alcohol intake [2] - or even to loss of consciousness [3]. E/I balance can be manipulated pharmacologically in various ways through drugs acting on glutamatergic and gamma-aminobutyric-acid (GABA)-ergic targets, but also through psychotomimetics like ketamine [4–6] and possibly psychedelics like psilocybin [7] and lysergic acid diethylamide [78].

Disturbances of E/I balance have featured in aetiological theories of numerous neurological and psychiatric disorders like epilepsy [9], autism [10], depression [11], and schizophrenia [12–14]. In the case of schizophrenia, for instance, multiple research lines point towards E/I imbalance, including genetic evidence implicating α-amino-3-hydroxy-5-methyl-4-isoxazolepropionic acid (AMPA) and N-methyl-D-aspartate (NMDA) receptors [1516], and postmortem studies that suggest reduced dendritic spine density and interneuron deficits [1718]. Additionally, electrophysiological signals measured with electroencephalography (EEG) and magnetoencephalography (MEG) are changed in individuals at clinical high risk for psychosis and patients with schizophrenia. This includes reduced gamma-band activity [19], mismatch negativity [20–24] and P300 in oddball paradigms [22232526], and reduced event-related potentials (ERPs) in paired-click paradigms [27–29]. There are pharmacological studies that reproduce ERP reductions observed in schizophrenia with ketamine and GABAergic drugs suggesting potential links with E/I balance [5630–37]. However, the field needs new methods to estimate E/I balance in vivo from non-invasive recordings to identify the precise nature of circuit dysfunction underlying these M/EEG biomarkers.

Here, we propose a canonical microcircuit model specifically designed to estimate E and I cell function from non-invasive electrophysiological recordings with dynamic causal modelling (DCM). Unlike fMRI-based models of E/I balance (e.g., [38]) that are limited by the hemodynamic response function acting as a low-pass filter focusing on M/EEG allows us to exploit rich temporal information M/EEG provides. We simulate how changes in E/I parameters can explain ERP alterations observed in schizophrenia to highlight their predictive validity, and we systematically assess how these parameters interact with each other through a series of simulations. The aims of this paper are threefold: (1) to introduce this new method for studying E/I balance, (2) to demonstrate the model’s utility for studying neuropsychiatric disorders using schizophrenia as an example, and (3) to provide a comprehensive resource for the scientific community by making the modelling code publicly available along with empirically-informed priors, for future investigations of E/I balance across various clinical conditions.

Methods

Canonical microcircuit for estimating E/I balance

Our model is grounded in the DCM framework. This state-space modelling framework was originally introduced to model fMRI data [3940] and later extended to M/EEG [41–44] based on seminal work from Jansen and Rit [4546]. This modelling approach can be concisely summarised by two sets of equations (in standard form of dynamical systems):1\documentclass[12pt]{minimal}
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\begin{document}$$\frac{{dx}}{{dt}}=f\left(x\left(t\right),u\left(t\right),{\rm{\theta }}\right)$$\end{document}dxdt=fxt,ut,θ2\documentclass[12pt]{minimal}
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\begin{document}$$y\left(t\right)=g\left(x\left(t\right),{\rm{\theta }}\right)+\epsilon$$\end{document}yt=gxt,θ+ϵ

The state equation (Eq. 1) describes how system states xx often simply referred to as “states” or “variables” in dynamic-systems notation (here voltage and current of neuronal populations) evolve in time as a function of states xx at time tt, a thalamic input into the system uu at time tt and model parameters \documentclass[12pt]{minimal}
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\begin{document}$${\rm{\theta }}$$\end{document}θ (e.g., expressing synaptic connectivity between neuronal populations). The observer equation (Eq. 2) describes how changes in the hidden states xx are mapped onto the measured signal yy at time tt (e.g., an EEG recording) through an observation model with parameters \documentclass[12pt]{minimal}
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\begin{document}$${\rm{\theta }}$$\end{document}θ and an observation error ϵ.

DCM for EEG comes in two forms: conductance-based and convolution-based models. The former models the membrane conductance of each neuronal population and specific ion channels (GABA, AMPA and NMDA receptors) based on Hodgkin and Huxley’s model of the squid giant axon [47] scaled from single cells to neuronal populations with the Morris and Lecar reduction [48–50]. Conversely, convolution-based DCM models are based on Jansen and Rit [4546] and model the conversion of incoming spikes to postsynaptic potentials (PSPs) through convolution with a synaptic response kernel [41–445152]. The E/I model we propose generalises across both convolution-based and conductance-based models, but we focus on the convolution-based model to introduce the E/I parameters. Next, we will unpack the modelling approach across different spatial scales.

Microscale model: neuronal population

At the microscale, we model a neuronal population through two basic operations (Fig. 1). First, at dendritic spines, incoming spikes are converted into PSPs through convolution with a synaptic response function:3\documentclass[12pt]{minimal}
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\begin{document}$$h\left(t\right)=\left\{\begin{array}{l}\displaystyle\frac{1}{\tau }t\,{e}^{-\displaystyle\frac{1}{\tau }t},\quad{t\ge 0}\\\quad\, 0,\quad\quad{t < 0}\end{array}\right.$$\end{document}h(t)={1τte−1τt,t≥00,t<0where τ is the synaptic time constant, with larger ττ leading to more prolonged responses in time (Fig. 1).

Secondly, PSPs \documentclass[12pt]{minimal}
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\begin{document}$${V}_{{in}}$$\end{document}Vin are converted into an average spike rate σ at the axon hillock using a sigmoid function:4\documentclass[12pt]{minimal}
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\begin{document}$${\rm{\sigma }}=\frac{1}{1+{e}^{-S\cdot {V}_{{in}}}}-{V}_{0}$$\end{document}σ=11+e−S⋅Vin−V0where \documentclass[12pt]{minimal}
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\begin{document}$$V_{0}$$\end{document}V0 the baseline firing rate, \documentclass[12pt]{minimal}
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\begin{document}$${V}_{{in}}$$\end{document}Vin is average membrane potential of the neuronal population, and parametrizes the slope of the sigmoid function and captures the dispersion of firing thresholds within this neuronal population, with smaller values of resulting in a shallower slope (i.e., more dispersed thresholds in the population, Fig. 1). Each population can be modelled through the convolution of these two functions:5\documentclass[12pt]{minimal}
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\begin{document}$$V=h\otimes {\rm{\sigma }}$$\end{document}V=h⊗σwhich results in a system of two coupled ordinary differential equations for each neuronal population, the first describing the change in voltage and the second describing the change in current:6\documentclass[12pt]{minimal}
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\begin{document}$$\begin{array}{c}\dot{v}=i\\ {\dot{i}}=f\left(\mathrm{input}\right)-\frac{2}{{\rm{\tau }}}\dot{v}-\frac{1}{{{\rm{\tau }}}^{2}}v\end{array}$$\end{document}v˙=ii˙=finput−2τv˙−1τ2v

The input is determined by intrinsic (within-source) and extrinsic (between-source) connectivity.

Mesoscale model: canonical microcircuit

At the mesoscale, we model a cortical column using the canonical microcircuit for predictive coding comprising inhibitory interneurons, spiny stellate cells, as well as superficial and deep pyramidal cells, based on previous work [53–56]. We connect these neuronal populations in the same cortical source based on Shaw et al. [57] (Fig. 1), which was previously applied to model data from patients with schizophrenia by Adams et al. [58] (for model comparisons with the previous model and model recovery analyses, see Supplement).

Importantly, we parameterise the self-connection of E (superficial pyramidal cells) and I cells (inhibitory interneurons), which we refer to as \documentclass[12pt]{minimal}
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\begin{document}$${g}_{{ee}}$$\end{document}gee and \documentclass[12pt]{minimal}
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\begin{document}$${g}_{{ii}}$$\end{document}gii, where the first subscript indicates the origin of the connection (ee for the excitatory and ii for the inhibitory population) and the second subscript specifies the target population. We focus on superficial pyramidal cells based on evidence suggesting that the excitability of superficial pyramidal cells is reduced in schizophrenia [58]. These parameters determine the excitability of E and I cells and change their average output firing rate. In this study, we operationalise E/I ratio as the ratio of average output firing rate of E and I cells that can be manipulated through changing these two parameters. Increasing \documentclass[12pt]{minimal}
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\begin{document}$${g}_{{ee}}$$\end{document}gee, for example, corresponds to increasing the connection strength of the (inhibitory) self-connection, which models a loss of E cell excitability (or reduced E/I ratio). Conversely, increasing \documentclass[12pt]{minimal}
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\begin{document}$${g}_{{ii}}$$\end{document}gii models a loss of I cell excitability resulting in net disinhibition of the microcircuit. Both self-connections are always inhibitory to ensure that the model generates stable dynamics. We can generalise this approach to the conductance-based model by parameterising the (self-inhibitory) GABAergic connections of the same neuronal populations.

We make two important modelling choices regarding these E/I parameters: (1) we constrain these parameters to be the same across all modelled regions in order to ensure an efficient global estimate of E and I cell excitability and (2) we allow them to be both general and condition-specific, expressing the average E/I balance across all conditions and the difference in E/I balance between conditions, respectively.

First, whilst there may be regional variation in E and I, we assume that genetic predisposition to altered E/I is generalised across cortex, rather than confined to specific areas (e.g., based on [59–62] in the case of schizophrenia-related changes). This is reflected in our cortex-wide parameter with the added potential benefit of using all the EEG data to estimate it. Note, however, that this assumption can be relaxed if one wishes to model gradients of E/I balance across the brain either by relaxing the global equality constraint or by adding a multiplicative weight of the global parameter (e.g., based on a receptor density map).

Second, we introduce these E and I parameters both as condition-specific parameters (capturing condition-specific rebalancing of E/I ratio in response to experimental inputs), referred to as \documentclass[12pt]{minimal}
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\begin{document}$${B}^{{g}_{{ii}}}$$\end{document}Bgii in the remainder of the paper, and condition-independent parameters expressing the average E/I balance, referred to as \documentclass[12pt]{minimal}
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\begin{document}$${g}_{{ii}}$$\end{document}gii. This is motivated by the notion that E/I changes are adaptive and may well have more impact on some EEG conditions, e.g. those that typically generate a much larger response (like oddballs). To summarise, the key differences between this model and Shaw et al.’s model [57] are (1) imposing equality constraints on intrinsic self-connections of E and I cells \documentclass[12pt]{minimal}
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\begin{document}$${g}_{{ii}}$$\end{document}gii and (2) introducing condition-specific modulation of self-connections of E and I cells \documentclass[12pt]{minimal}
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\begin{document}$${B}^{{g}_{{ii}}}$$\end{document}Bgii.

Macroscale model: network

Lastly, we connect different sources with excitatory forward connections \documentclass[12pt]{minimal}
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\begin{document}$${A}^{F}$$\end{document}AF that originate from superficial pyramidal cells in one source and project to spiny stellate cells and deep pyramidal cells of another source, and inhibitory backward connections \documentclass[12pt]{minimal}
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\begin{document}$${A}^{B}$$\end{document}AB which originate from deep pyramidal cell populations and project to superficial pyramidal cells and inhibitory interneurons [5556]. This results in the following system of coupled differential equations for each source. Note that these coupled equations have the form of Eq. (6), but for consistency with previous work and the provided code, we denote all hidden states (voltage and current) with xx. For spiny stellate cells, we have:7\documentclass[12pt]{minimal}
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\begin{document}$$\begin{array}{c}\dot{{x}_{1}}={x}_{2}\\ \dot{{x}_{2}}=\frac{1}{{{\rm{\tau }}}_{{ss}}}\left({A}^{F}\cdot {\rm{\sigma }}\left({x}_{3}\right)-{g}_{2}\cdot {\rm{\sigma }}\left({x}_{5}\right)-{g}_{1}\cdot {\rm{\sigma }}\left({x}_{1}\right)+U-2\dot{{x}_{2}}-\frac{{x}_{1}}{{{\rm{\tau }}}_{{ss}}}\right)\end{array}$$\end{document}x1˙=x2x2˙=1τssAF⋅σx3−g2⋅σx5−g1⋅σx1+U−2x2˙−x1τss

For E cells (superficial pyramidal cells), we have:8\documentclass[12pt]{minimal}
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\begin{document}$$\begin{array}{c}\dot{{x}_{3}}={x}_{4}\\ \dot{{x}_{4}}=\frac{1}{{{\rm{\tau }}}_{{sp}}}\left(-{A}^{B}\cdot {\rm{\sigma }}\left({x}_{7}\right)-{g}_{11}\cdot {\rm{\sigma }}\left({x}_{5}\right)+{g}_{7}\cdot {\rm{\sigma }}\left({x}_{1}\right)-{g}_{6}\cdot {\rm{\sigma }}\left({x}_{3}\right)-2\dot{{x}_{4}}-\frac{{x}_{3}}{{{\rm{\tau }}}_{{sp}}}\right)\end{array}$$\end{document}x3˙=x4x4˙=1τsp−AB⋅σx7−g11⋅σx5+g7⋅σx1−g6⋅σx3−2x4˙−x3τsp

For I cells (inhibitory interneurons), we have:9\documentclass[12pt]{minimal}
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\begin{document}$$\begin{array}{c}\dot{{x}_{5}}={x}_{6}\\ \dot{{x}_{6}}=\frac{1}{{{\rm{\tau }}}_{{ii}}}\left(-{A}^{B}\cdot {\rm{\sigma }}\left({x}_{7}\right)-{g}_{3}\cdot {\rm{\sigma }}\left({x}_{5}\right)+{g}_{4}\cdot {\rm{\sigma }}\left({x}_{1}\right)+{g}_{10}\cdot {\rm{\sigma }}\left({x}_{3}\right)+{g}_{5}\cdot {\rm{\sigma }}\left({x}_{7}\right)-2\dot{{x}_{6}}-\frac{{x}_{5}}{{{\rm{\tau }}}_{{ii}}}\right)\end{array}$$\end{document}x5˙=x6x6˙=1τii−AB⋅σx7−g3⋅σx5+g4⋅σx1+g10⋅σx3+g5⋅σx7−2x6˙−x5τii

And lastly for deep pyramidal cells, we have:10\documentclass[12pt]{minimal}
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\begin{document}$$\begin{array}{c}\dot{{x}_{7}}={x}_{8}\\ \dot{{x}_{8}}=\frac{1}{{{\rm{\tau }}}_{{dp}}}\left({A}^{F}\cdot {\rm{\sigma }}\left({x}_{3}\right)-{g}_{8}\cdot {\rm{\sigma }}\left({x}_{5}\right)+{g}_{12}\cdot {\rm{\sigma }}\left({x}_{3}\right)-{g}_{9}\cdot {\rm{\sigma }}\left({x}_{7}\right)-2\dot{{x}_{8}}-\frac{{x}_{7}}{{{\rm{\tau }}}_{{dp}}}\right)\end{array}$$\end{document}x7˙=x8x8˙=1τdpAF⋅σx3−g8⋅σx5+g12⋅σx3−g9⋅σx7−2x8˙−x7τdp

We implemented this model using SPM12 (version 7771, Wellcome Trust Centre for Neuroimaging, London, UK, https://github.com/spm/spm12/releases/tag/r7771) in MATLAB (version R2020b; The MathWorks Inc., Natick, MA, USA; https://www.mathworks.com). The code for both the convolution- and conductance-based E/I models can be found on GitHub (https://github.com/daniel-hauke/dcm_ei).

Simulations

To illustrate the effects of the E and I parameters, we conduct a series of simulations based on parameter estimates from data collected from a large number of healthy volunteers during three different paradigms as part of the Bipolar and Schizophrenia Network for Intermediate Phenotypes (B-SNIP) [6263] and the second North American Prodrome Longitudinal Study (NAPLS2) [64–66]. We briefly recapitulate the most pertinent information about the paradigms and preprocessing below. For more detailed information, please see the referenced studies and the Supplement.

Participants

We analysed EEG data from n = 266 (paired-click paradigm), n = 241 (passive oddball paradigm), and n = 237 (active oddball paradigm) healthy participants. The EEG setup and preprocessing is reported in the Supplement. To provide priors for future analyses – that can be used beyond investigations focused on schizophrenia – we computed the grand mean data from healthy control participants. We ground our simulations in these denoised grand mean ‘participants’ to illustrate the effects of changing E and I parameters.

Tasks

Paired-click paradigm

Participants were presented with 150 binaural broadband auditory click pairs (4 ms, 75 dB, inter-click interval: 500 ms, average interstimulus interval: 9.5 s) [67].

Passive oddball paradigm

Participants performed a visual oddball primary task while being presented with 1794 tones that were either standard tones (85%,633 Hz, 50 ms), duration deviants (5%, 633 Hz, 100 ms), pitch deviants (5%, 1000 Hz, 50 ms) or duration+pitch double-deviants (5%, 1000 Hz, 100 ms). Tone length included 5 ms rise/fall time. Subjects were told to ignore the tones and focus on the visual task. Here, we focused on modelling the double-deviant tone as it showed the largest clinical effects in terms of predicting conversion to psychosis in individuals at clinical high risk for psychosis [66].

Active oddball paradigm

In the active oddball paradigm, participants were presented with 450 stimuli including 80% standard tones (500 Hz, 50 ms with 5 ms rise/fall time),10% target tones (1000 Hz, 50 ms, 5 ms rise/fall time) and 10% novel sounds with a stimulus onset asynchrony of 1250ms [65]. Novel sounds were various natural and man-made sounds [68]. Participants were asked to respond to target tones by pressing a button.

Networks

We defined three task-specific networks to be modelled with our E/I model.

Paired-click paradigm

Since we are unaware of previous studies that used DCM to model paired-click responses, we defined a task network based on previous literature [6970] comprising left (MNI: [−53 −20 13]) and right superior temporal gyrus (STG; [49 −10 10]) as auditory input regions, left ([−36 23 3]) and right insular cortex (Ins; [36 23 3]), and left ([−1 −20 67]) and right medial frontal gyrus (MFG; [1 −20 67]). Note that primary auditory cortex (A1) is modelled implicitly through the exogenous input to limit the number of model parameters. To confirm that these sources explain sufficient variance in the data, we conducted a multiple sparse prior (MSP) source analysis with 16 mm spheres drawn around these coordinates and compared it to an (unconstrained) independent and identically distributed (IID) source analysis in SPM. These six sources explained 95.06% of the variance versus 97.18% in the IID case. Formal model comparison based on the model evidence suggested that there was strong evidence that the 6-region source reconstruction was the better model (log Bayes factor: 23.6) [71].

Passive oddball paradigm

We modelled the passive oddball paradigm with a well-established 6-region network comprising bilateral A1 (left: [−42 −22 7], right: [46 14 8]) as input regions, STG (left: [−61 −32 8], right: [59 −25 8]) and inferior frontal gyrus (IFG; left: [−46 20 8], right: [46 20 8]) based on extensive previous work [2172–77] including various DCM studies [5878–80].

Active oddball paradigm

We defined a 6-region network for the active oddball paradigm based on a previous meta-analysis of 75 oddball paradigm PET/fMRI studies [81]. We chose STG as input regions (left: [−61 −32 8], right: [59 −25 8]) using the same coordinates as for the passive oddball task network and confirmed that they fell into the brain region activated during the active oddball task [81]. Importantly, in contrast to the passive oddball paradigm, the active oddball paradigm requires participants to maintain top-down attentional control towards the deviant tones to respond with button presses [82]. We thus modelled bilateral inferior frontal junction (IFJ; left: [−56 7 29], right: [50 8 30]) and intraparietal sulcus (IPS; left: [−33 −42 64], right: [33 −42 64]) regions of the dorsal attention network based on Kim [81]. We found that this 6-region model explained 94.31% of the variance versus 95.14% of an IID source reconstruction and was strongly favoured in model comparison (log Bayes factor: 49.2).

Integration

We integrated the convolution-based model with a Euler integrator [83], which we found to have the best accuracy-runtime trade-off for our purposes (see [83] for a runtime comparison). We also provide code that works with the default SPM12 integrators and an implementation using MATLABs dde23 integrator for benchmarking the conductance-based model (Supplement).

Priors

We used the default SPM12 priors with the following changes. In the source model, we implemented equality constraints on transmission delays (extrinsic and intrinsic) and intrinsic connectivity parameters across regions to drastically reduce the number of free parameters by 40 (20 → 2 delay and 24 → 2 intrinsic connectivity parameters) for a six-region model. We used the equivalent current dipole (ECD) forward model and assumed that only E cells (superficial pyramidal cells) contributed to the EEG signal (fixing an additional 2 parameters), although note that this assumption can be easily relaxed.

The neuronal time constants ττ of the original canonical microcircuit model were selected to model gamma oscillations [56]. We found that, under the new integration schemes, the default priors on ττ were not suitable for generating the oscillations that we wished to model – especially for the active oddball task, where the characteristic responses are on slower (theta) time scales. In simulations, we also observed that the slope of the sigmoid output firing rate function SS — parametrising the dispersion of firing thresholds or depolarisation of a neuronal population [84] — was crucial in ensuring that the model generated dampened as opposed to ongoing or increasing oscillations. We thus performed a grid search over different combinations of the four time constants and S spanning the following grids \documentclass[12pt]{minimal}
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\begin{document}$${\tau }_{{dp}}$$\end{document}τdp: [2 16 32 64 128] ms and SS: [−2 −1 0 1 2]. We identified the best model based on the free energy approximation to the log model evidence, while constraining the solutions to physiological plausible parameter ranges, which we defined as [1 60] ms for spiny stellate cells [85–88], [1 30] ms for inhibitory interneurons [8990] and [1 200] ms for pyramidal cells. The τ range for pyramidal cells was motivated by evidence that NMDA receptors on these neurons can have time constants of around 40–200 ms [9192], and these receptors may be captured implicitly in the convolution-based model. All priors are summarised in Table 1.Table 1Parameter Priors.

Parameter Description PriorMean Prior Precision

Neuronal Source Model f(x)

A{1} Average extrinsic connectivity (forward, SP to SS) 200 1/16

A{2} Average extrinsic connectivity (forward, SP to DP) 25 1/16

A{3} Average extrinsic connectivity (backward, DP to SP) 50 1/16

A{4} Average extrinsic connectivity (backward, DP to II) 100 1/16

B Condition-specific extrinsic connectivity 0 1/8

C Input Strength 32 1/32

G Average intrinsic connectivity [gee gii] [800 800] 1/32

BG Condition-specific intrinsic connectivity [Bgee Bgii] 0 1/8

T Synaptic time constant [SS SP II DP] Paired Click: [16 16 2 16] ms

Passive oddball: [16 32 2 2] ms

Active oddball: [16 128 2 2] ms

1/32

D Neuronal delays (all sources) [intrinsic extrinsic] [1 8] ms 1/64

S Slope of output firing rate function Paired Click: e−2

Passive oddball: e−1

Active oddball: e−1

1/64

R Stimulus [onset dispersion] Paired Click: [50 30] ms

Passive oddball: [68 16] ms

Active oddball: [100 16] ms

1/1024

Forward Model g(x)

Lpos Source location [MNI coordinates] Paired Click: [−53 −20 13;

49 −10 10;

−36 23 3;

36 23 3;

−1 −20 67;

1 −20 67]

Passive oddball: [−42 −22 7;

46 −14 8;

−61 −32 8;

59 −25 8;

−46 20 8;

46 20 8]

Active oddball: [−61 −32 8;

59 −25 8;

−56 7 29;

50 8 30;

−33 −42 64;

33 −42 64]

0

L Dipole orientation 0 64

J Neuronal populations contributing to the EEG

[SS SP II DP]

[0 1 0 0] 0

Note, that the grid search identified different priors for the slope of the sigmoid function SS and the synaptic time constants TT for different paradigms (cf. Priors section). The optimal values are determined by the overall duration of the time window that is modelled and the dominant frequencies; for example, more alpha in paired-click and passive oddball paradigms compared to more theta in the active oddball paradigm. The input parameters were selected to ensure that the input peak coincides with the first dominant peak of the event-related potential of each paradigm. For the motivation of the three different networks (Lpos), please see the Networks section. DP deep pyramidal cells. II inhibitory interneurons. SP superficial pyramidal cells. SS spiny stellate cells.

Model diagnostics

To assess model fit (i.e., accuracy), we computed the Pearson correlation between the responses predicted by the model and the measured EEG responses. To assess parameter identifiability, we fit the model to 100 randomly selected healthy participants using the posteriors from the grand mean as empirical priors. We report the Pearson correlation between E/I parameters across participants to test whether parameters are identifiable (i.e., their posterior dependencies or correlations are small). In line with previous work [93], we consider r < |0.6| to be an acceptable correlation, suggesting that the parameters mediate distinct processes. To measure parameter recovery, we (i) simulated synthetic data using the posterior parameters from models fit to 100 randomly selected participants, (ii) added empirical levels of noise, estimated from the data (i.e., using the posterior estimate saved in DCM.Ce), and (iii) reinverted the models on the simulated data. We report intraclass correlation coefficients (ICC), as recommended previously [9495], and consider recovery to be poor (<0.4), fair (0.4–0.59), good (0.6–0.74), or excellent (>0.75) as suggested by Fleiss [96]. We also report Pearson correlations between simulated and recovered parameters in line with previous work [9397].

Parameter sensitivity analysis

To illustrate the effects of the E and I parameters, we simulated ERPs starting with their posterior and incrementally increased or decreased the parameters over the interval [−0.5 0.5] in steps of 0.125, which corresponds to scaling the connectivity strength of the self-connection by ±50%, while keeping all other parameters fixed to their posterior estimate. We also varied multiple parameters together. To simplify the visualisation for multidimensional simulations, we display the amplitude of characteristic difference wave ERP components. Specifically, we defined the P2 component as the most positive deflection in Cz between 120–250 ms in the paired-click paradigm, the MMN, as the most negative deflection in Fz between 150–250 ms in the passive oddball paradigm, and the P3b component as the most positive deflection in Pz between 250–600 ms in the active oddball paradigm.

Results

Model diagnostics

The model fit all paradigms remarkably well. Specifically, it predicted responses in the paired stimulus paradigm with an average r = 0.95 (tone 1: r = 0.95; tone 2: r = 0.96; Fig. 2a), in the passive oddball paradigm with r = 0.89 (standard: r = 0.85; deviant: r = 0.94; Fig. 2d) and the active oddball paradigm with r = 0.84 (standard: r = 0.71; target: r = 0.97; Fig. 2g). Note that ERPs tend to be much larger in the deviant and target conditions of the oddball paradigm compared to the standard tone condition, while the noise is comparable between conditions. This results in reduced SNR in the standard condition and likely explains why the model shows reduced fits for this condition. We observed good parameter identifiability (all r < |0.6| across all paradigms; Fig. 2b, e and h). Parameter recovery of \documentclass[12pt]{minimal}
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\begin{document}$${B}^{{g}_{{ee}}}$$\end{document}Bgee was excellent, while \documentclass[12pt]{minimal}
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\begin{document}$${B}^{{g}_{{ii}}}$$\end{document}Bgii recovery was fair across all three paradigms (Fig. 2c, f, i) and comparable to other findings [98].

Posterior parameter estimates

Posterior parameter estimates are reported in Supplementary Tables 1–3 and provided in SPM12 DCM format on GitHub (https://github.com/daniel-hauke/dcm_ei). We recommend using these parameters as empirical priors for future DCM analyses with this model and these paradigms.

Simulating changes in E and I cell excitability

To illustrate the validity of the model, we simulated a loss of condition-specific E or I cell excitability across all three paradigms and compared those results to known effects in the schizophrenia literature (Fig. 3). We found that increasing \documentclass[12pt]{minimal}
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\begin{document}$${B}^{{g}_{{ee}}}$$\end{document}Bgee to simulate a loss of pyramidal cell excitability reduced N1 and P2 components of the difference waveform in the paired click paradigm by reducing the response to the first tone and increasing the response to the second tone (Fig. 3a), which may explain reduced S2/S1 ratios observed in patients [27–29]. Another finding in the schizophrenia literature is that patients show reduced responses to both tones [27], which can be simulated with our model by increasing \documentclass[12pt]{minimal}
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\begin{document}$${g}_{{ee}}$$\end{document}gee (reducing average excitability across both conditions, Fig. S1). Interestingly, simulating a loss of inhibitory cell excitability by increasing \documentclass[12pt]{minimal}
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\begin{document}$${B}^{{g}_{{ii}}}$$\end{document}Bgii had no effect on the waveform, suggesting that this parameter alone cannot explain the changes observed in schizophrenia (Fig. 1b). Increasing I-to-E coupling (\documentclass[12pt]{minimal}
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\begin{document}$${g}_{11}$$\end{document}g11) did not change this result. Note, however, that the effect of changing \documentclass[12pt]{minimal}
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\begin{document}$${B}^{{g}_{{ii}}}$$\end{document}Bgii depends on the other parameters (see Fig. S3 for illustrations of single participants).

In the passive oddball paradigm, we found that a loss of E cell excitability (↑\documentclass[12pt]{minimal}
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\begin{document}$${B}^{{g}_{{ee}}}$$\end{document}Bgee) resulted in reduced ERP amplitudes in responses to deviant tones and an overall reduced difference waveform with reduced mismatch negativity and P3a responses (Fig. 3c), both observed in schizophrenia [2024–266566]. A similar change in I cell excitability resulted in no comparable change in the ERP (Fig. 3d) nor did it when I-to-E coupling (\documentclass[12pt]{minimal}
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\begin{document}$${g}_{11}$$\end{document}g11) was increased.

For the active oddball paradigm, we observed distinct effects of changing E and I cell excitability. While reduced E/I ratio - either through a loss of E cell excitability (↑\documentclass[12pt]{minimal}
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\begin{document}$${B}^{{g}_{{ee}}}$$\end{document}Bgee) or hyperexcitable I cells (↓\documentclass[12pt]{minimal}
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\begin{document}$${B}^{{g}_{{ii}}}$$\end{document}Bgii) - resulted in reduced P3b in the target-standard difference waveform (Fig. 3e), changing E cell excitability affected the ERPs in response to both standard and target tones, whereas changing I cell excitability only affected the response to target tones (Fig. 3f). To investigate whether this qualitative distinction would also hold in the passive oddball paradigm, we increased the simulation range for I cell excitability to ±200% (not shown) and indeed observed similar changes (i.e., much stronger effects on the deviant- compared to the standard-ERP). This finding is in line with P3b reductions observed in schizophrenia patients [2526] and suggests that the model can potentially distinguish E and I cell dysfunction because they relate to different data features.

Interactions between E and I cell excitability

Next, we investigated whether E and I cell excitability interacted, by systematically varying both parameters together. We found that the effect of changing E cell excitability on the ERP amplitude was largely independent of varying I cell excitability in the paired-click paradigm, as indicated by the straight vertical line in Fig. 4a, which suggests that changing E cell excitability has the same effect on the P2 component irrespective of assumed I cell excitability. We observed a similar result for the MMN component in the passive oddball paradigm (Fig. 4b). However, we found that this was primarily driven by the effect of changing E cell excitability on the deviant response for all levels of I cell excitability. There was no effect on the standard-ERP. Lastly, we found non-trivial interactions for the P3b component: reducing the E/I ratio through simulating both a loss of E cell excitability (↑\documentclass[12pt]{minimal}
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\begin{document}$${B}^{{g}_{{ee}}}$$\end{document}Bgee) and an increase in I cell excitability (↓\documentclass[12pt]{minimal}
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\begin{document}$${B}^{{g}_{{ii}}}$$\end{document}Bgii) resulted in the most pronounced effects on P3b components of the target-ERP and the difference waveform, whereas the effects of changing E cell excitability on the standard-ERP were largely independent of I cell excitability (Fig. 4c). This suggests that P3b amplitudes could reflect changes in both parameters rather than E or I cell function alone (see Supplement for more detailed investigations of parameter interactions for this paradigm).

Discussion

This study set out to introduce a canonical microcircuit model to estimate E/I balance from non-invasive M/EEG recordings by parameterising global excitatory and inhibitory cell function. We illustrated how this model can be used to study condition-specific changes in E/I balance by applying it to three different auditory paradigms: paired-click, active oddball and passive oddball. We found that E/I parameters can be recovered efficiently using a variety of empirical paradigms in healthy participants. We then illustrated— with numerical studies—that a loss of pyramidal cell excitability can account for many known empirical findings in the schizophrenia M/EEG literature, including reduced ERP amplitudes across all three paradigms. Lastly, we provide the requisite code, integration schemes and empirical priors for future analyses of data from clinical cohorts.

We found excellent parameter recovery for the parameter capturing pyramidal cell excitability, rendering it a promising and most importantly, biologically interpretable biomarker. This parameter could capture multiple biological processes that change pyramidal cell excitability, for example, NMDA receptor function on pyramidal cells as suggested previously [58], or neuromodulatory effects or loss of synapses within the superficial pyramidal cell pool. This parameter should thus be interpreted with caution and future studies are needed to carefully evaluate the contributions of these processes to the E/I parameters. Possible avenues for validation include pharmacological studies administering GABAergic and glutamatergic agents, magnetic resonance spectroscopy studies in humans, human-derived induced pluripotent stem cell (iPSC) models and animal studies using opto- and chemogenetic manipulations. Alternatively, conductance-based DCMs can be used, which contain explicit parameterisations of NMDA receptors and thus may assess their function more directly; these models can be more challenging to fit, however. Interestingly, our simulations suggest that a loss of pyramidal cell excitability may be sufficient to explain the effects observed in patients with schizophrenia across three different auditory paradigms, namely reduced responses in the paired-click, active oddball and passive oddball paradigms [2024–296566].

Secondly, our simulation (Fig. 2c) and model comparison (Fig. S4d) analyses showed that changing a single model parameter is sufficient to explain changes in the MMN. This provides a very parsimonious explanation without sacrificing biological plausibility compared to previous models, which entail changes in many different parameters including local excitability as well as extrinsic forward, backward and cross-hemispheric connectivity [80]. Our model is still in line with predictive coding accounts of the MMN [2199], which view the MMN as a precision-weighted prediction error signal that is changed in psychosis [100101]. A key feature of our model is that it enables the characterisation of E/I balance, predictive coding and their relationship within the same model, unlike other models that have focused on one view or the other, emphasizing the predictive coding view like classic canonical microcircuit models [55–58] or the E/I balance view by modelling only E and I populations [438] including Wilson-Cownan-type models [102]. In classic predictive coding accounts [103], forward connections are thought to convey prediction errors and backward connections to convey predictions. Synaptic gain can be viewed as a mechanism instantiating precision-weighting of prediction errors and predictions [21], and in our model, determining E/I balance. Reducing E/I balance through the parameters we introduced can thus be viewed as reducing the precision of prediction errors or lowering the learning rate in canonical Bayesian models of belief updating (e.g., [104105]) — a mechanism that would result in increased weighting prior information, which has been considered as an explanation for hallucination in schizophrenia [106107]. Note, however, that other biological mechanisms can implement precision-weighting, for example sharper population tuning. Future research is needed to determine when this mechanism should be included in the model.

Our model also includes deep pyramidal cells, which are thought to send predictions through extrinsic backward connections [55]. The excitability of these cells can be parameterised by adding another global E parameter to their self-inhibition and relaxing the prior on their contribution to the M/EEG signal in the future. This would allow testing whether the excitability of superficial and deep pyramidal cells differently relates to precision-weighted prediction errors and predictions, respectively.

According to the predictive coding view, changing pyramidal cell excitability corresponds to changing the precision-weighting or learning rate used to scale prediction errors elicited by deviant tones. Thus, our results suggest that a failure of precision-weighting can explain reduced ERPs across all three paradigms. Importantly, this scaling likely involves NMDA receptor-related processing [58], because cortical gain parameters scale or modulate synaptic connections multiplicatively [108109] as opposed to additively unlike other cellular processes such as spatial and temporal summation in the cell soma. The contributions of AMPA, GABA and NMDA receptors can be further explored in future studies with the conductance-based version of the canonical microcircuit we provide here.

Beyond schizophrenia, we believe this model can be used to study E/I balance alterations across various other clinical conditions and the effect of pharmacological interventions on E/I. For example, it is well suited for understanding changes in E/I balance in depression [11] and the effects of fast-acting antidepressant effects observed following ketamine administration [4110] by modelling suitable M/EEG biomarkers like the MMN. Another interesting avenue for future research will be to apply this model to understand how E/I balance is altered under the influence of psychedelics. Leptourgos et al. [7] recently proposed that psychedelic-induced hallucinations are likely caused by different circuit changes compared to hallucinations associated with schizophrenia. It has been suggested that 5-HT2A receptor activation on pyramidal cells may lead to disinhibition of these cells [111112]. Our model could be used to test this theory empirically and one may expect to find increased excitability of pyramidal cells following psychedelic administration, which would capture 5-HT2A-mediated disinhibition. Additionally, both predictive coding accounts [113] and E/I balance accounts of psychedelics [78] have been proposed and can be investigated together. Our model can be used to determine whether the excitability of superficial pyramidal cells associated with prediction errors or deep pyramidal cells sending back predictions are changed under psychedelics as proposed previously [7113]. In the future, it will also be important to study E/I imbalance in other disorders including epilepsy [9114–116], autism [10] and Alzheimer’s [117118].

There are several limitations of this study that merit attention. First, the recovery for the inhibitory cell excitability parameter was only fair. While this suggests the parameters can in principle be recovered from these paradigms, its recovery will need to be improved, especially for clinical applications. Future work should investigate whether other paradigms like the 40 Hz auditory steady state paradigm [19] or resting-state are better suited for probing inhibitory cell function. Secondly, it will be important to investigate the test-retest reliability of E/I parameters, because our parameter recovery results only provide an upper bound on the reliability of this computational assay of E/I balance [94]. Third, since we parameterise E cell excitability through an (inhibitory) self-connection, it is also possible that this parameter reflects a local inhibitory loop perhaps involving parvalbumin-positive (PV + ) interneurons. This will need to be clarified in animal studies or iPSC models. iPSCs will also be important to inform priors and constrain parameters (e.g., synaptic time constants) to biologically plausible values in the future.

In conclusion, we introduced a canonical microcircuit for estimating E/I balance. Our simulations suggest that a loss of pyramidal cell excitability may account for various ERP changes observed in schizophrenia. We hope that this tool can serve as a computational assay to assess cell and receptor function based on non-invasive M/EEG or optically pumped magnetometers (OPM) recordings [119120], enabling the study of E/I balance across various clinical conditions and ultimately permitting allocation of different treatments to individual patients to restore E/I balance and alleviate symptoms.

Supplementary information

References

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