Impaired long-range excitatory time scale predicts abnormal neural oscillations and cognitive deficits in Alzheimer’s disease

Alzheimer’s disease (AD) is the most common form of dementia, progressively impairing memory and cognition. While neuroimaging studies have revealed functional abnormalities in AD, how these relate to aberrant neuronal circuit mechanisms remains unclear. Using magnetoencephalography imaging we documented abnormal local neural synchrony patterns in patients with AD. To identify abnormal biophysical mechanisms underlying these abnormal electrophysiological patterns, we estimated the parameters of a spectral graph-theory model (SGM). SGM is an analytic model that describes how long-range fiber projections in the brain mediate the excitatory and inhibitory activity of local neuronal subpopulations. The long-range excitatory time scale was associated with greater deficits in global cognition and was able to distinguish AD patients from controls with high accuracy. These results demonstrate that long-range excitatory time scale of neuronal activity, despite being a global measure, is a key determinant in the spatiospectral signatures and cognition in AD.

Notation All the vectors and matrices are written in boldface and the scalars are written in normal font.The frequency f of a signal is specified in Hertz (Hz) and the corresponding angular frequency ω = 2πf is used to obtain the Fourier transforms.The connectivity matrix is defined as C = c jk , where c jk is the connectivity strength between regions j and k, normalized by the row degree.

Mesoscopic model
For every region k out of the total N regions, we model the local excitatory signal x e (t), local inhibitory signal x i (t) as well as the long range excitatory signal x k (t) where the global connections are incorporated.The local signals x e (t) and x i (t) are the same for every region k.They are modeled using an analytical and linearized form of neural mass equations.We write a set of differential equations for evolution of x e (t) and x i (t) due to decay of individual signals with a fixed neural gain, incoming signals from coupled excitatory and inhibitory signals, and input white Gaussian noise.Letting f e (t) and f i (t) denote the ensemble average neural impulse response functions, the x e (t) and x i (t) are modeled as where, ⋆ stands for convolution, p(t) is input noise, parameters g ee , g ii , g ei are neural gain terms, and parameters τ e , τ i are characteristic time constants.These are global parameters and are the same for every region k.Here, the ensemble average neural impulse response functions f e (t) and f i (t) are assumed to be Gamma-shaped and written as , and,

Macroscopic model
A similar equation is written for the macroscopic signal x k (t), for every k th region, accounting for long-range excitatory cortico-cortical connections for the pyramidal cells.The evolution of x k (t) is assumed as a sum of decay due to individual signals with a fixed excitatory neural gain, incoming signals from all other connected regions determined by the white matter connections, and the input signal x e (t) + x i (t) determined from Eq. ( 2) and (3).Signal x k is modeled as where, τ G is the graph characteristic time constant, α is the global coupling constant, c jk are elements of the connectivity matrix determined from DTI followed by tractography, τ v jk is the delay in signals reaching from the j th to the k th region, v is the cortico-cortical fiber conduction speed with which the signals are transmitted.The delay τ v jk is calculated as d jk /v, where d jk is the distance between regions j and k.This set of equations is parameterized by 8 global parameters: excitatory time constant τ e , inhibitory time constant τ i , macroscopic graph time constant τ G , excitatory neural gain g ee , inhibitory neural gain g ii , coupled population neural gain g ei , global coupling constant α, and conduction speed v.The neural gain g ee is kept as 1 to ensure parameter identifiability.We estimate the remaining 7 global parameters using an optimization procedure described in the next section.

Model solution in the Fourier domain
Since the above equations are linear, we can obtain a closed-form solution in the Fourier domain as demonstrated below.The Fourier transform F() is taken at angular frequency ω which is equal to 2πf , where f is the frequency in Hz.The Fourier transform of the equations ( 2) and (3) will be the following, where F(x e (t)) = X e (ω) and F(x i (t)) = X i (ω), and j is the imaginary unit: τ e (g ee X e (ω) − g ei F i (ω)X i (ω)) + P (ω), and, (6) Here, P (ω) is the Fourier transform of the input Gaussian noise p(t) which we assume to be identically distributed for both the excitatory and inhibitory local populations for each region, and the Fourier transforms of the ensemble average neural response functions are On solving the above equations ( 6) and ( 7), X e (ω) and X i (ω) are where, , and, F 3 = jω + g ee τ e F e (ω) . (10) Then, the transfer functions H e (ω) and H i (ω) can be separated out and we get X e (ω) = H e (ω)P (ω) , and X i (ω) = H i (ω)P (ω) .
In order to obtain a Fourier response of the macroscopic signal, we first re-write Eq. ( 5) in the vector form We similarly take a Fourier response of the macroscopic signal and obtain the following as the Fourier transform of Eq. ( 13), where F(x(t)) = X(ω): where, Note that each element in the matrix C is normalized already by the row degree.The above equation can be re-arranged to give Here, we define the complex Laplacian matrix: where, I is the identity matrix of size N × N .The eigen-decomposition of this complex Laplacian matrix is where, U (ω) are the eigenmodes/eigenvectors and Λ(ω) = diag([λ 1 (ω), . . ., λ N (ω)]) consist of the eigenvalues λ 1 (ω), . . ., λ N (ω), at angular frequency ω.The macroscopic response X(ω) from Eq. ( 15) becomes By using the eigen-decomposition of the Laplacian matrix, this yields where, u k (ω) are the eigenmodes from U (ω) and λ k (ω) are the eigenvalues from Λ(ω) obtained by the eigen-decomposition of the complex Laplacian matrix L(ω) obtained in Eq. 17. Equation ( 19) is the closedform steady state solution of the macroscopic signals at a specific angular frequency ω.We use this modeled spectra to compare against empirical MEG spectra and subsequently estimate model parameters.Overall workflow is shown in Fig. S1.The structural connectome and the distance matrix are shown in Fig. S5 and S6, respectively.

Model performance
Fig. S2 shows the regional and band-specific correlations.Fig. S2A shows the regional spectral correlations for healthy controls (top) and patients with AD (bottom).The spectral correlations are greater than 0.5 for most subjects and regions.Precisely, the spectral correlations were greater than 0.5 for more than 90% and 93% of controls and AD, respectively.Fig. S2B shows the spatial correlations for the delta-theta, alpha, and beta frequency bands for healthy controls (left) and patients with AD (right).The spatial correlations are also greater than 0.5 for most subjects and frequency bands.Precisely, the spatial correlations were greater than 0.5 for more than 94% and 97% of controls and AD, respectively.

Figure S1 :
Figure S1: SGM models excitatory and inhibitory neuronal subpopulation signals that influence the longrange excitatory signals.The long-range signals are connected to each other via the structural connectome, and these signals transmit with a fixed conduction speed.SGM provides a closed-form solution in the frequency domain.This is compared to the frequency spectra obtained from MEG for inferring the SGM model parameters.

Figure S2 :
Figure S2: A: Regional spatial correlations for healthy controls (top) and patients with AD (bottom).Every point indicates the spectral correlation for a specific subject in a brain region indexed in the x-axis.B: Spatial correlations for healthy controls (left) and patients with AD (right).Every point indicates the spatial correlation for a specific subject in a frequency band indexed in the x-axis.

Figure S3 :
Figure S3: Empirical (left) and SGM (right) PSD for control (top) and AD (bottom) for specific subjects with spectral correlation approximately equal to the mean correlation.The spectral correlation for control was 0.72 and for AD was 0.78 for the data shown here.

Figure S4 :
Figure S4: Spatial pattern of the spectral power in the delta-theta frequency band for subjects with spatial correlation in the delta-theta band approximately equal to the mean correlation.Top for controls and bottom for AD.Left for MEG and right for SGM.The minimum and maximum values were based on the dynamic range for each of them.The spatial correlation for control was 0.63 and for AD was 0.67 for the data shown here.