Model of auditory cortex
We use a computational model of AC introduced by May and colleagues \citep{May2013,May2015,Westoe2016} with the aim of linking non-invasive results with in-vivo single- and multiunit observations. The intuition behind this previous modelling work has been that many auditory phenomena emerge from large-scale interactions in the auditory cortex. With detailed information lacking on the human AC, the model borrows the core-belt-parabelt organization of the primate AC \citep{Baumann2013,Hackett2014}, as shown in Figure \ref{fig:AC_w_ee}A, with multiple streams of feedforward and feedback activation indicated by the arrows.
A central feature of the model is short-term plasticity of the synapses. The model explains a wide range of temporal binding (across-time) phenomena observed in invasive experiments experiments in terms of synaptic plasticity and anatomical structure. These phenomena include: forward masking \citep{Brosch1997,Brosch2008}, stimulus-specific adaptation \citep{Ulanovsky2003,Ulanovsky2004}, two-tone facilitation \citep{Brosch1999}, and selective responses to complex sounds such as speech and monkey calls \citep{Rauschecker1997}. Further phenomena explained by the model include the adaptation of the N1(m), and the emergence of the mismatch response as a dependence of the N1(m) on stimulus statistics \citep{May2010}. We note that by including as much of the anatomy of AC as possible at the cost of keeping the local dynamics as simple as possible, this approach diverges from modelling efforts which describe a single core field (usually primary AC, A1) and are intended to explain a limited aspect of auditory processing, such as frequency tuning and forward masking \citep{Loebel2007}, auditory induction in rat AC \citep{Noto2016}, bird song discrimination \citep{Larson2009}, or the N1/N1m response in humans \citep{Wang2013}.
The dynamical unit of the model is the cortical column comprising two state variables representing the population of excitatory (pyramidal) neurons and inhibitory interneurons.
This is the mean-field leaky-integrator model of neural dynamics as formulated by \citet{Hopfield1986} as a variation of the classic \citet{Wilson1972} model of interacting neural populations. The Hopfield and Tank formulation is slightly closer to the compartmental model, and the state variables can be seen as an approximation of the membrane potential whose time derivative depends on the cross-membrane currents. The output of the dynamical unit is the spiking rate, which is a continuous, non-linear function of the membrane potential. While originally intended as a single-unit description, the leaky-integrator can be used as a population description by assuming that the units in the population are identically and symmetrically connected with each other, and that they all receive the same external input.11This could be mistakable. The external input only goes to core columns. In this case the population units behave identically with each other, and the population can be described by the unit equation. The equations referring to cross-membrane currents (i.e. synaptic and leak currents) make it easier to motivate the calculation of the EEG and MEG signal, as will be seen below. As shown in Section 3, the leaky-integrator formulation also has the advantage that it opens up an analytical approach to the system dynamics.
Central to the model is the anatomical structure of AC (Figure \ref{fig:AC_w_ee}A) which is captured in the weight matrices \(W_{\tiny\mbox{ee}},W_{\tiny\mbox{ei}}\), and \(W_{\tiny\mbox{ie}}\) (Figure \ref{fig:AC_w_ee}B). The AC has a core-belt-parabelt organization which is similar across species22There is a CBP organization across species, but I would not call it similar (gerbil: 2-3-3; monkey: 3-8-2). \citep{Budinger2000,incollectionBudinger2005,Baumann2013,Hackett2014}. In general, core fields are characterized by short latency on-responses to pure tones and their preferential connections with the tonotopically organized division of the auditory thalamus. They have extensive local connections with each other and with the surrounding belt fields. Belt fields are also tonotopically organized, albeit with a lesser spatial frequency resolution. There are strong local connections of belt fields with core fields and neighbouring parabelt fields as well as connections with other cortical areas. In addition to dense connections with the ventral division of the medial geniculate body, belt fields also have pronounced connections with non-tonotopic parts of the auditory thalamus.33These connections, however, are not captured by the model, and neither are those reported for the parabelt in the following sentence. Parabelt fields are non-tonotopic and isocortical, with lower cell density than the belt fields, and have connections mainly with non-tonotopic auditory and non-auditory thalamic nuclei and remote cortical areas.
Our model replicates this structure with its 224 columns (208 cortical, 16 subcortical) being evenly distributed into one tonotopically organized thalamic field, three core fields, eight belt fields, and two parabelt fields where each field comprises 16 columns. We also include a single, tonotopically organized area representing the inferior colliculus (IC) and feeding afferent input to the thalamus.
As shown in Figure \ref{fig:AC_w_ee}A, the areas of the model are connected according to the scheme found in the macaque \citep{Kaas2000}; the corresponding connection matrices are displayed in Figure \ref{fig:AC_w_ee}B. We assume that all inter-column connections are excitatory and encapsulated in the matrices \(W_{\tiny\mbox{ie}}\) and \(W_{\tiny\mbox{ee}}\); the latter also includes the long-range connections between fields. Further, we assume that \(W_{\tiny\mbox{ie}}\) only has intra-field elements, and thus functional inhibition is of the lateral type. The matrix \(W_{\tiny\mbox{ei}}\) only has diagonal elements and describes inhibitory-to-excitatory connections. This connectivity scheme was used in the previous May et al. modelling work, which also included short-term synaptic plasticity. Here, we exclude synaptic plasticity for analytical expediency, with the intention of addressing this theme in future work. We also note that the omission of plasticity does not affect the validity of the current results.
In the current study, we restrict ourselves to the MEG response produced by the model. The MEG was assumed to be directly proportional to the synaptic input to the excitatory populations, summed over the columns. This approximation of the primary current by the input current is justified by theoretical considerations \citep{May1999} and simulation results \citep{inproceedingsMay2002} using realistic neuron models in the NEURON simulation environment \citep{Hines1997} (or: \citet{Hines2001}).