Criticality in Alzheimer’s and healthy brains: insights from phase-ordering

Abstract

Criticality, observed during second-order phase transitions, is an emergent phenomenon. The brain operates near criticality where complex systems exhibit high correlations. As a system approaches criticality, it develops “domain”-like regions with competing phases and increased spatio-temporal correlations that diverge. The dynamics of these domains depend on the system’s proximity to criticality. This study explores the differences in the proximity to criticality of Alzheimer’s-afflicted and cognitively normal brains through the use of a spin–lattice model derived from resting-state fMRI data and investigates the type of criticality found in the human brain - whether it is of the Ising class or something more complex. The temporal correlations in both groups display a stretched exponential nature, indicating closer alignment with the criticality of the spin-glass class rather than the Ising class. Longer relaxation times observed in cognitively normal subjects suggest increased proximity to the phase boundary. The weak distinction observed in the spatial characteristics related to proximity to criticality might once more point to a spin-glass scenario, necessitating nuanced order parameters to distinguish between phase-ordering in Alzheimer’s and cognitively normal brains.

Introduction

The brain is a highly complex system with a very large number of interdependent parts that exhibit non-linearity and emergent collective behavior very similar to models of criticality near a phase transition (Chialvo 2010) such as the Ising model studied in statistical physics. Functional magnetic resonance imaging (fMRI) (Glover 2011) is a primary tool used to study brain activity by signals sensitive to blood flow and oxygenation at a local area of the brain. fMRI uses the “hemodynamic response” to measure regional activity in the brain (Iadecola 2017). The signal detected by fMRI is not a direct measure of neural activity; however, it is an indirect measure of the hemodynamic response to neural activity. By measuring changes in blood flow and oxygenation, fMRI can provide a spatial and temporal map of brain activity, allowing one to identify which regions of the brain are active during different cognitive or perceptual tasks.

The Ising model (Baxter 2016) is a physical model commonly used to understand the brain. The model represents a lattice with spins located at each lattice point, which is described by the Hamiltonian $H=-{\sum }_{<ij>}J{S}_i{S}_j$, where $S_i$ and $S_j$ represent the spins at lattice points i and j and J is the constant pairwise coupling between these points. The spins can take values of “+1” or “-1”, and the summation is limited to its nearest neighbors (as denoted by $<>$). This Hamiltonian captures the energetic aspects, while the entropy arises from the collective degrees of freedom of the spins.

Criticality marks the boundary of phases when the transition is second order. The brain has been shown to display critical behavior operating at a state between order and disorder (Beggs and Plenz 2003, 2004; Linkenkaer-Hansen et al. 2001) and featuring scale-invariant activity patterns (Novikov et al. 1997). Chialvo (Chialvo 2010) conjectures that criticality is a crucial aspect of the learning and memory capacity of the brain. A brain that is sub-critical can be seen as an equilibrium state that is too simplistic to learn and respond effectively, while a brain that is critical has long-range correlations and small fluctuations that can bring about global changes in the neuronal patterns, which makes it a good learning system but poor in memory capacity. It is likely that the brain exists or tunes itself between these two regimes to achieve optimal efficiency. Clinical relevance to brain criticality has been an area of intense research (Zimmern 2020).

In the Ising model, at low temperatures, the energetics of the system dominate over entropy, resulting in spin alignment and the phase known as the ferromagnetic phase. Conversely, at high temperatures, entropy dominates, and the spins exhibit random behavior, which is referred to as the paramagnetic phase. The Ising model demonstrates predictable collective behavior and undergoes a second-order transition at a critical temperature $T_c$ between these two phases. As one approaches $T_c$, competing “domains” emerge due to the interplay between thermal fluctuations and spin interactions (Chaikin et al. 1995). Domains are regions within the material where the magnetization is uniform and distinct from its neighboring regions. The resting-state brain has been shown (Eguiluz et al. 2005; Fox et al. 2005) to display correlated and anticorrelated subnetworks which are dynamic and spatially distributed, precisely the signature of domains in spin models. Domain formation in a spin–lattice model can be studied to characterize criticality through their spatial and temporal correlation lengths.

As the temperature approaches $T_c$, the Ising spins exhibit critical behavior characterized by power-law scaling, for example, the magnetic susceptibility diverges as $\chi \sim |T-T_c{|}^{-\gamma }$, where $\gamma$ is the critical exponent for the magnetic susceptibility. Similarly, the correlation length diverges as $\xi \sim |T-T_c{|}^{-\nu }$, where $\nu$ is the critical exponent for the correlation length. The criticality observed in the brain is known to be “self-organized” (Das et al. 2014; Tognoli and Kelso 2014; Walter and Hinterberger 2022), while the Ising model is a model that is commonly used to understand criticality. At $T_c$, the system displays scale-invariant behavior, characterized by fluctuations existing across all length scales. This behavior is described by a general scale-invariant mechanism.

The criticality of the Ising model is of interest because it serves as a simplified model for comprehending complex systems that exhibit similar behavior, such as the brain (Chialvo 2010). In the brain, neurons interact with one another, and the nature of their interaction depends on their activity levels. Like the Ising model, the brain can display critical behaviour (Beggs and Plenz 2003, 2004; Linkenkaer-Hansen et al. 2001), featuring scale-invariant activity patterns (Novikov et al. 1997). However, brain activity patterns are not limited to nearest neighbors, and the interactions are not uniform. A variant of the Ising model, the Sherrington-Kirkpatrick model (SK) (Sherrington and Kirkpatrick 1975), allows for interactions beyond nearest neighbors and incorporates a distribution of coupling strengths, similar to the interaction of fMRI signals between non-local voxels. The SK variant introduces additional physics, including a new phase known as the spin-glass phase. In this phase, the spins experience frustration and exhibit glassy behaviour (Mezard et al. 1987). Frustrated regions in spin-glass phases do not reach equilibrium easily, allowing for long-term memory which is not seen in the Ising model. The Hamiltonian for the SK model is defined as $H=-\frac{1}{2}{\sum }_{\mathit{ij}}{J}_{\mathit{ij}}{S}_i{S}_j$. Here, i and j are any two spins on the lattice, N is the number of spins, and $J_{\mathit{ij}}$ are the interaction strengths taken from a Normal distribution $P(J_{\mathit{ij}})\sim exp(-N{J}_{\mathit{ij}}^2/2J^2)$, with J the characteristic value of the spin-spin interaction. The strength of spin interactions $J_{\mathit{ij}}$ can be calculated using a maximum likelihood approach based on binary activity patterns derived from fMRI data. Since the maximum likelihood is an entropy maximum state, the distribution of $J_{\mathit{ij}}$ around the maximum likelihood is a Normal distribution, while the distribution of $J_{\mathit{ij}}$ found in the Ising model is a delta function.

If we were to map fMRI signals onto a spin–lattice model, drawing an analogy between voxels and binarized spins, we could in principle discern the nature of the inherent criticality of the system (without a priori assuming Ising or SK models) by studying critical indicators. The distinction that arises in the critical indicators due to proximity to criticality will differ in the case of Ising-like criticality compared to spin-glass-like criticality, allowing us to liken the criticality found in the brain to either of these.

Alzheimer’s disease (AD) is a neurodegenerative disorder that affects memory and cognition. Since cognition relies on the production and synchronization of neuronal signals (Breakspear and Terry 2002), studying the collective behavior of these signals is an appropriate way to investigate AD. Previous studies have explored whether AD exhibits deviations from criticality. In normal individuals, synchronization in electroencephalography (EEG) shows power-law scaling (Stam and De Bruin 2004). Power-law behavior is also observed in individuals with AD, but with decreased amounts in certain frequency regimes compared to non-demented patients (Stam et al. 2005). The power-law exponents of the spectral densities showed statistically significant differences between AD and control subjects in the temporal and frontal lobes (Vysata et al. 2014), which is consistent with frontal lobe atrophy associated with AD. Magnetoencephalography (MEG) studies, which infer magnetic fields produced by brain electric currents, showed decreased autocorrelations and oscillation bursts in the signals compared to controls (Montez et al. 2009). These studies suggest that Alzheimer’s disease tangibly affects proximity to criticality.

The motivation of our work is to explore brain criticality through phase-ordering analysis of fMRI signals mapped to a spin–lattice model. We quantitatively investigate phase-ordering using spatial and temporal metrics in Alzheimer’s and cognitively normal brains. Our focus lies in discerning the variations within these metrics to classify the kind of underlying criticality. Using phase-ordering statistics, we can ascertain the nature of criticality - be it Ising-like or spin-glass.

Materials and methods

Data acquisition

The study utilized resting-state functional magnetic resonance imaging (rs-fMRI) data from the Alzheimer’s Disease Neuroimaging Initiative (ADNI) database (Petersen et al. 2010). To obtain rs-fMRI data for Alzheimer’s patients (AD) and cognitively normal (CN) subjects, we used the portal: https://adni.loni.usc.edu/. We found 121 AD scans and 243 CN scans on the portal. The ADNI dataset contains multiple rs-fMRI scans for each subject, ranging from a single scan per subject to five scans per subject. To prevent bias in the data toward subjects with multiple scans, a single rs-fMRI scan was selected for each subject from the earliest available date. The corresponding anatomical scans were also extracted from the same dates for use in preprocessing. Our final data set consisted of rs-fMRI scans for 89 subjects, including 34 AD (M:F = 16:18, mean age $=73.18\pm 7.28$ years) and 55 CN subjects (M:F = 25:30, mean age $=76.91\pm 6.31$ years).

Preprocessing

The fMRI data was preprocessed primarily using tools from the FMRIB Software Library (FSL) (Jenkinson et al. 2012; Smith et al. 2004). First, motion correction was performed using FSL’s MCFLIRT (Jenkinson et al. 2002) to align all volumes to the mean volume, producing motion parameters and mean images as output. Next, FSL’s SliceTimer was used for slice-timing correction. The coregistration step included the following procedures: (1) skull-stripping the anatomical image using FSL’s BET, (2) segmenting it with FSL’s FAST and thresholding the resulting white matter probability image, (3) pre-alignment and coregistration of the fMRI to the anatomical images using FSL’s FLIRT, and (4) applying the computed coregistration transformation to the functional and mean images. The images were spatially smoothed using SPM with a full-width at half-maximum (FWHM) of 5 mm. Nipype’s ArtifactDetect algorithm (Gorgolewski et al. 2011) was used to detect and separate out artifacts from the functional images, with a norm threshold of 2 and z-intensity threshold of 3. Finally, Nilearn (Abraham et al. 2014) was used to calculate and apply a brain mask, which utilized the histogram of the mean fMRI image intensity and discarded the bottom 20% and top 15% of it.

Domain formation

To map rs-fMRI data onto the spin–lattice model, we assigned each voxel in the fMRI volume a unique 3D lattice point. For each voxel, the fMRI signal values were utilized to determine spin values. Let $\mathbf{x}$ represent the time series of the fMRI signal within a given voxel, of length $K=T_{\text{max}}$ (where $T_{\text{max}}$ is the length of the time-series). Let $\mathbf{z}$ be a sorted array of the values of $\mathbf{x}$, i.e, $z_1\ge z_2\ge \dots \ge z_K$. We then compute the average maximum, $Z_{\text{max}}$, and average minimum, $Z_{\text{min}}$, values using:

$$\begin{array}{cc}\hfill Z_{\text{max}}=& \frac{1}{10}\sum _{i=1}^{10}{z}_{(i)}\hfill \\ \hfill Z_{\text{min}}=& \frac{1}{10}\sum _{i=1}^{10}{z}_{(K-i+1)}\hfill \end{array}$$

$Z_{\text{max}}$ is the mean of the 10 largest values and $Z_{\text{min}}$ is the mean of the 10 smallest values. The signal $\mathbf{x}$ is then converted to the binarized series of spins $\mathbf{s}=\{s_1,s_2,\dots ,s_K\}$ as follows:

$$\begin{array}{c}\hfill s_i=\left\{\begin{array}{cc}1,\hfill & x_i\ge (Z_{\text{max}}+Z_{\text{min}})/2\hfill \\ -1,\hfill & \text{otherwise}\hfill \end{array}\right)\end{array}$$ 1

Thus, the fMRI signal from voxels is transformed into a spin–lattice for each time point. Note, the threshold for each signal is sometimes taken to be the mean of the signal, however, this restricts the spin model to the paramagnetic phase. The above method for calculating the threshold avoids this issue. Finally, by connecting neighboring spins on the lattice that have the same spin with an edge, we define a ‘domain’ as the resulting set of lattice points connected by these edges.

To compare AD and CN cases quantitatively, we identify domains formed throughout the time series. The number of domains ($n_{\mathit{dom}}$) and size of domains ($S_{\mathit{dom}}$) were computed for the entirety of the data. To identify domains assigned to the voxels across the signal time series for each subject, we use the Hoshen-Kopelman algorithm (Hoshen and Kopelman 1976). The distributions of domain sizes and number of domains can now be compared and analyzed.

Self-averaging

Self-averaging is a fundamental concept in thermodynamics, indicating that the statistics of a system improve with an increase in system size. According to the central limit theorem (CLT), fluctuations become proportional to $N^{-1/2}$, where N is the system size. However, the CLT assumes independence in the random variables whose average is being calculated. As a system nears criticality, spatial correlations typically increase, leading to a loss of this independence. Independent regions with distinct disorders may emerge, resulting in a breakdown of self-averaging. When averaging encompasses a region large enough to account for all disorders, it defines scales below which self-averaging deteriorates. We investigate the system scale where self-averaging breaks down to determine a minimum system size for reliable averaging of signals.

To investigate self-averaging, we define a quantity known as relative variance: $R_X=(\mathrm{\Delta }V)/{[X]}^2$, which depends on system size N. Here, X is a random variable taken from a distribution P(X), $\mathrm{\Delta }V=[X^2]-{[X]}^2$ is the variance of X, and [] denotes the average over realisations. The fMRI data for a subject is taken at a selected time point and converted into a flattened array of voxel strengths. Systems of different sizes are created from this fMRI data array by randomly selecting a fixed number ($M=30$) of non-overlapping subarrays of size N, and the mean of each subarray is calculated. These mean values then become the array values for the new system of size M. The $R_X$ value of each new system is then calculated. This process is repeated for system sizes $N=1$ to $N=1000$, and for all subjects at a fixed time t.

Time correlation

The auto-correlation function (ACF), which is denoted as $\rho (t)$, of the BOLD signal is calculated using the inverse Fourier transform power spectral density (PSD) and ACF as defined by Wiener-Khinchin theorem (Chatfield 1989). $\rho (t)$ is normalized by the value at $t=0$. A stretched exponential function of the form given below was fit to the ACF using the least squares method.

$$\begin{array}{c}\hfill {\rho }_s(t)=Aexp\left[-,{\left(\frac{t}{\tau }\right)}^{\beta }\right]+B\end{array}$$ 2

A is chosen to be unity and B is chosen to be the average value of the second half of the time-series. Finally, the relaxation time $\tau$ and the stretching parameter $\beta$ are extracted from the fit.

Results and discussion

We investigate the time correlation function of the fMRI time-series generated from each voxel in order to study the relaxation behavior of the signals. The power spectrum is calculated for each time series using the Wiener-Kinchin theorem (Chatfield 1989), which allows for the direct calculation of the auto-correlation through a straightforward Fourier transform of the power spectra (see Materials and methods).

Fig. 1(A–C) shows the extracted time correlation function, $\rho (t)$. Notably, $\rho (t)$ exhibits a better fit to a stretched exponential function. Stretched exponential relaxation, also known as the Kohlsrauch-Williams-Watts stretched exponential form (Williams and Watts 1970), involves a two-step relaxation pattern observed in the SK model (Billoire and Campbell 2011) and is a classic signature of metastable states approaching the glass transition.

Fig. 1.

A, B, C Time correlation $\rho (t)$. The time correlations fit stretched exponential functions ${\rho }_s(t)=Aexp[-{(t/\tau )}^{\beta }]+B$ (with relaxation time denoted by subscript ‘st-exp’) better than pure exponentials (subscript ‘exp’)

To accurately calculate the relaxation time, $\tau$, we fit $\rho (t)$ to both an exponential decay function and a stretched exponential decay function (${\rho }_{s(t)}$) using the least squares method. Figure 2 displays the distribution of $\beta$ and $\tau$ values calculated in the following manner:

$$\begin{array}{cc}\hfill P(\beta )& =\frac{1}{N_{\mathit{sub}}{N}_{\mathit{vox}}}\sum _i^{N_{\mathit{sub}}}\sum _j^{N_{\mathit{vox}}}\delta \left(\beta ,-,{\beta }_j^i\right)\hfill \end{array}$$ 3

$$\begin{array}{cc}\hfill P(\tau )& =\frac{1}{N_{\mathit{sub}}{N}_{\mathit{vox}}}\sum _i^{N_{\mathit{sub}}}\sum _j^{N_{\mathit{vox}}}\delta \left(\tau ,-,{\tau }_j^i\right)\hfill \end{array}$$ 4

Here, $N_{\mathit{sub}}$ is the total number of subjects, $N_{\mathit{vox}}$ is the total number of voxels, ${\beta }_j^i$ and ${\tau }_j^i$ are the values of $\beta$ and $\tau$ for $j^{\mathit{th}}$ voxel and $i^{\mathit{th}}$ subject. We observe that approximately 59% of voxels have $\beta <0.95$, indicating a deviation from exponential decay. This is a classical temporal signature of a spin-glass. AD and CN show no significant differences in $P(\beta )$. However, $P(\tau )$ shows (Fig. 2(B)) difference between AD and CN only in the large $\tau$ range ($\tau >80$ ). The inset in Fig. 2(A, B) shows a different way of averaging, specifically averaging over the voxels

$$\begin{array}{cc}\hfill P({\beta }^{\mathit{vox}})& =\frac{1}{N_{\mathit{sub}}}\sum _i^{N_{\mathit{sub}}}\delta \left(\beta ,-,\frac{1}{N_{\mathit{vox}}},\sum _j^{N_{\mathit{vox}}},{\beta }_j^i\right)\hfill \end{array}$$ 5

$$\begin{array}{cc}\hfill P({\tau }^{\mathit{vox}})& =\frac{1}{N_{\mathit{sub}}}\sum _i^{N_{\mathit{sub}}}\delta \left(\tau ,-,\frac{1}{N_{\mathit{vox}}},\sum _j^{N_{\mathit{vox}}},{\tau }_j^i\right)\hfill \end{array}$$ 6

$P({\beta }^{\mathit{vox}})$ shows that the entire distribution follows a stretched exponential pattern, and there are no significant differences between AD and CN cases. On the other hand, $P({\tau }^{\mathit{vox}})$ exhibits a clear distinction between the AD and CN cases, indicating that CN is closer to the critical temperature than AD. The presence of a stretched exponential relaxation suggests that some parts of the lattice may be in a spin-glass phase, contributing to the increased complexity of criticality compared to the Ising model (Mezard et al. 1987).

Fig. 2.

A Distribution of all $\beta$ values for the stretched exponential fit and (inset) $\beta$ values averaged over subjects for each voxel. B Distribution of all $\tau$ values and (inset) $\tau$ values averaged over subjects for each voxel

Self-averaging, or the lack thereof, can be quantified (Aharony and Harris 1996; Mezard et al. 1987; Roland and Grant 1989). Let us consider a random system where an observable property takes on random variable X taken from a distribution, P(X) with variance, $\mathrm{\Delta }V=[X^2]-{[X]}^2$, and its average, [X] (averaged over realizations of the randomness). The relative variance of the system is $R_X=\mathrm{\Delta }V/{[X]}^2$. According to the CLT, when $R_X\sim N^{-1}$, we say the system is self-averaging. However, when $R_X\sim N^{-\alpha }$ and $0<\alpha <1$, self-averaging is poor.

To examine self-averaging, we calculate $R_X$ for fMRI data in the CN and AD cases. Figure 3(A) shows the log-log plot of $R_X$ vs. N. $R_X$ exhibits two distinct slopes, transitioning from $\alpha =-0.424$ to ${\alpha }^{\ast }=-0.850$ at $N^{\ast }\approx 372$ for CN and from $\alpha =-0.435$ to ${\alpha }^{\ast }=-0.682$ at $N^{\ast }\approx 291$ for AD. $N^{\ast }$ is the intersection point between the linear fits for the first $50\%$ (slope $\alpha$) and last $50\%$ (slope ${\alpha }^{\ast }$) of the data points. The magnitude of the slope of the last $50\%$ for AD (${\alpha }^{\ast }=-0.682$) is significantly lower than that of CN (${\alpha }^{\ast }=-0.850$) implying worse self-averaging in the AD case. For comparison, we also plot $R_X$ for a random system with values taken from a Normal distribution with an $\alpha \approx -1$. The poor self-averaging observed indicates criticality in the presence of disorder. Fig. 3(C) shows the distribution of ${\alpha }^{\prime }$ (which is the slope of the fit on the entirety of the log-log plot of $R_X$ vs N) for individual subjects. We note a shift in the peak of the distribution towards the left and a broader distribution for CN compared to AD, implying marginally better self-averaging in the case of CN. $N^{\ast }$ depends on how close the system is to criticality. The significance of $N^{\ast }$ becomes apparent when statistics involve voxel averaging. In fMRI studies, due to the substantial spatial resolution of the signals, it is common to reduce voxel-wise data to a few hundred regions of interest (ROIs) based on pre-existing atlases (Roland and Zilles 1994). This is done through a process called parcellation where each voxel is mapped to an existing anatomical or functional parcel/ROI. The time series is obtained by averaging over the voxels present within a parcel. Our analyses indicate that parcellation with ROIs smaller than $N^{\ast }$ would lead to inadequate self-averaging. From the case studied here, the lower limit for the number of voxels in an ROI seems to be around 400. Through the self-averaging analysis, we observe poorer self-averaging in the case of AD compared to CN.

Fig. 3.

A Log-log plot of $R_X$ vs N is shown for mean over the control (CN), Alzheimer (AD) subjects, and random variables taken from a Normal distribution (Norm). The scatter points are mean values over subjects and the shaded regions show the standard deviation. The dashed lines represent a fit on the first $50\%$ of data points, and the dotted lines are fit on the final $50\%$ of the data. Their intersection point (fluorescent) $N_{\mathit{CN}}^{\ast }\approx 372$, $N_{\mathit{AD}}^{\ast }\approx 291$ marks N after which system moves closer to self-averaging. B Log-log plot of S(f) for CN and AD. Dashed lines show the mean linear fit and shaded regions show the standard deviation of S(f). C As $R_X\sim N^{\prime }$, the distribution of ${\alpha }^{\prime }$: $P({\alpha }^{\prime })$ derived from (A) is shown for the CN and AD cases. D The distribution m, the slope of S(f)

Further, we calculate the power spectrum S(f) for brain signals versus frequency (f) in the AD and CN cases, as shown in Fig. 3(B). The power spectrum S(f) follows a power-law relationship $S(f)\propto f^m$, where the exponent m characterizes the color of the noise. A power spectrum that follows $S(f)\propto 1/f$ indicates self-similarity and modular hierarchical organization in the brain (Expert et al. 2011). We observe the 1/f behavior in both AD and CN cases, with mean exponents ranging from $-0.98\pm 0.45$ for AD to $-1.00\pm 0.44$ for CN. The distribution of m is depicted in Fig. 3(D) and is similar for both AD and CN, suggesting that the hierarchical self-similar organization may not differ significantly between the two cases.

We then investigate the domains present in fMRI signals, which we analyze by associating with a spin system after binarization (see Materials and methods for details). The domains formed throughout the time series are computed for all subjects. Domains are represented by black and white regions in Fig. 4, where white indicates +1 spins and black indicates -1 spins. Our analysis reveals the presence of large percolating domains of magnetization with +1 and -1 spins in both the AD and CN cases. The existence of such large domains suggests that the system is close to the critical temperature ($T_c$) for the subjects studied (Chaikin et al. 1995). To study domain formation quantitatively, the number of domains - $n_{\mathit{dom}}$ - and the size of the domains - $S_{\mathit{dom}}$ - are computed for all the subjects. We compute the distributions $P(n_{\mathit{dom}})$ and $P(S_{\mathit{dom}})$ throughout the time-series of length $T_{\mathit{max}}$, considering all voxels and all subjects:

$$\begin{array}{cc}\hfill P(S_{\mathit{dom}})& =\frac{1}{N_{\mathit{sub}}{T}_{\mathit{max}}}\sum _j^{N_{\mathit{sub}}}\sum _k^{T_{\mathit{max}}}\delta (S_{\mathit{dom}}-S^j(k))\hfill \end{array}$$ 7

$$\begin{array}{cc}\hfill P(n_{\mathit{dom}})& =\frac{1}{T_{\mathit{max}}}\sum _k^{T_{\mathit{max}}}\delta \left(n_{\mathit{dom}},-,\frac{1}{N_{\mathit{sub}}},\sum _j^{N_{\mathit{sub}}},n^j,(k)\right)\hfill \end{array}$$ 8

Here, $n^j(k)$ and $S^j(k)$ represent the number of domains and size at time k for subject j. Figure 5(A) shows the $S_{\mathit{dom}}$ distribution for large domains ($S_{\mathit{dom}}>500,2000$). In Fig. 5(B) we find that smaller domains ($S_{\mathit{dom}}<50$) are dominated by sizes $⪅$ 10 domains. Since the distribution is very sparse, relevant domain size limits were set to study the large-domain region (in (A) and (C)) and the small-domain region (in (B) and (D)). There appear to be no significant differences in cluster sizes between AD and CN for both small and large domains. Our investigation shows large, percolating domains in the time series for every subject. This can be seen in the peak in (A) at $S_{\mathit{dom}}\approx 15000$, corresponding to the peaks at $n_{\mathit{dom}}\approx 2.0$ and $n_{\mathit{dom}}\approx 2.4$ in (C). Interestingly, there also seem to be small, isolated domains which can be seen in the initial peak in (B), corresponding to the peak at $n_{\mathit{dom}}\approx 300$ in (D). This seems to imply the existence of a couple of stable clusters with the dynamics mostly revolving around smaller domains.

Fig. 4.

Domain formation in near critical brain of Alzheimer’s (AD) and control (CN) subjects for $t=15\text{s}$ and $t=150\text{s}$

Fig. 5.

A, B Domain size distributions $P(n_{\mathit{dom}})$ for all subjects throughout the time-series limited to domains of size (A) $S_{\mathit{dom}}>500,2000$ and (B) $0<S_{\mathit{dom}}<50$. C, D Distribution of the number of domains $P(n_{\mathit{dom}})$ averaged over all subjects (at each time point) limited to domains of size (C) $S_{\mathit{dom}}>500,2000$ and (D) $S_{\mathit{dom}}>0$. E Distribution of the mean time of largest domains, ${\tau }_{\mathit{ld}}$ (in seconds) averaged over all subjects

Furthermore, we track the largest domain $S_{\mathit{ld}}(t)$ for each fMRI scan over time by finding the domain that has maximal overlap with the largest domain at the previous time point. This is done over all time points. We then calculate the autocorrelation function $<S_{\mathit{ld}}(t)S_{\mathit{ld}}(0)>$ for each subject. The autocorrelation shows an exponential decay. We obtain relaxation times ${\tau }_{\mathit{ld}}$ using an exponential fit. Figure 5(E) displays the distribution of relaxation times for the AD and CN subjects. We observe that the relaxations also exhibit an exponential decay, and the distribution of ${\tau }_{\mathit{ld}}$ is similar for both AD and CN. No significant differences are found in the domain sizes and populations, as well as the lifetimes of the largest domains when comparing AD to CN. We see a significant distinction between AD and CN in the case of temporal correlation, but not in spatial correlation. This leads us to believe that the criticality found in the brain cannot be captured by simple Ising order parameters.

Conclusion

We have conducted an investigation into the phase-ordering domains and critical dynamics of Alzheimer’s disease (AD) and cognitively normal (CN) individuals. We observed that both AD and CN exhibit time correlation functions with stretched exponential features, similar to those found in the spin-glass phase. From the distribution of mean relaxation times, we observe a shift in the tail of the CN distribution towards larger values. At the critical temperature ($T_c$), the relaxation time tends to diverge, so a longer relaxation time indicates a closer proximity of CN subjects to $T_c$. However, this does not seem to hold true in terms of domain sizes, as the domain sizes of CN are not significantly larger than those of AD. We see a clear distinction in the temporal correlation lengths but not in the spatial correlation lengths. This again leads us to believe that criticality in the brain may not be as straightforward as the criticality observed in the Ising model, as both temporal and spatial correlation lengths are expected to diverge in the Ising model with increasing proximity to criticality. However, such a clear distinction is not expected in the case of spin-glass-like criticality as the “frozen” characteristics of spin-glass systems are more accurately described by temporal measures rather than spatial measures.

Spin-glass criticality can account for the lack of a significant distinction in the spatial correlation lengths as local frustrations might not allow for the system to reach equilibrium domain distributions. The hypothesis suggesting that the brain operates as a near-critical system, capable of fine-tuning its criticality for optimal learning and memory (Chialvo 2010), is founded upon an inherent Ising-like criticality assumption. However, the relatively simple criticality within the Ising class lacks the capacity for long-term, enduring memory-an attribute naturally embedded in the SK model within the spin-glass phase. In this phase, localized frustrations counteracting thermodynamic equilibrium can lead to sustained memory, evident in the stretched exponential behavior of temporal correlations. We have investigated the breakdown of self-averaging and identified the scales beyond which self-averaging holds in the fMRI lattice model which is a quenched disordered SK model. These scales have been ignored in earlier works and they should be weighed in any statistical analysis of this kind. Our findings indicate that the nature of criticality in the brain is more likely of the spin-glass type (Ezaki et al. 2020), offering a broader range of complex features to explore (Mezard et al. 1987). The observed absence of substantial and definitive distinctions in phase-ordering between individuals with Alzheimer’s disease (AD) and cognitively normal (CN) individuals underscores the necessity for a deeper inquiry. Our preliminary findings also prompt further investigation into the assertions regarding the brain’s efficiency and its distal properties from criticality (O’Byrne and Jerbi 2022).

Our manuscript’s significance lies in pointing to the inadequacy of simplistic Ising-like order parameters in effectively distinguishing between spin-glass states. This challenge is reminiscent of difficulties encountered in glass physics, where the spatial signatures of glassy states often defy differentiation despite exhibiting relaxation times with an order of magnitude of separation. This analogy aptly underscores the complexity of distinguishing between AD and CN subjects solely based on order relaxation times, given the intricate nature of their disparities. Using order parameters that depend on higher-order correlations like the 4-point correlation function in glassy systems can potentially discern subtler spatial differences in glassy liquids not too deep into the glassy phase, which is not employed in our investigation. The availability of data with a definitive gradation of closeness to criticality would allow for a more thorough investigation into signatures of spin-glass criticality through higher-order correlation parameters. We leave these investigations for future perusal.

Our specific identification of criticality akin to spin-glass behavior paves the way for the creation of statistical brain models that integrate long-term memory, a feature that the traditional Ising model does not to support. Furthermore, this recognition serves to establish a bridge between neuroscience research and the extensive physics literature in the realm of frustrated systems.

Data Availability

The data used in this work is available at https://adni.loni.usc.edu/. Further details regarding the data are mentioned in the Data Acquisition subsection under Materials and Methods within the manuscript.

Declarations

Conflict of interest

The authors have no competing interests to declare that are relevant to the content of this article.