Brain signaling becomes less integrated and more segregated with age

The integration-segregation framework is a popular first step to understand brain dynamics because it simplifies brain dynamics into two states based on global vs. local signaling patterns. However, there is no consensus for how to best define what the two states look like. Here, we map integration and segregation to order and disorder states from the Ising model in physics to calculate state probabilities, Pint and Pseg, from functional MRI data. We find that integration/segregation decreases/increases with age across three databases, and changes are consistent with weakened connection strength among regions rather than topological connectivity based on structural and diffusion MRI data.


INTRODUCTION
Aging is the number one risk factor for almost all neurodegenerative diseases (Kennedy et al., 2014).For every 5 years after the age of 65, the probability of acquiring Alzheimer's disease doubles (Bermejo-Pareja et al., 2008).An influential conceptual approach to begin making sense of brain dynamics frames it in terms of a balance between integrated and segregated network states (Deco, Tononi, Boly, & Kringelbach, 2015;Friston, 2009;Sporns, 2010Sporns, , 2013;;Tononi, Sporns, & Edelman, 1994;Wig, 2017).On one hand, the brain faces functional pressure to have as many regions directly connected for quick communication.On the other hand, the brain is constrained to minimize metabolic energy consumption because it consumes ten-times more of the body's energy than expected by mass (Raichle, 2006).Tuning the balance between extensive global signaling, referred to as integration, and limited local signaling, referred to as segregation, optimally compromises between functional and energetic constraints (Bullmore & Sporns, 2012;Cohen & D'Esposito, 2016;Manza et al., 2020;Wang et al., 2021).Although these constraints remain throughout life, aging disrupts their balance.
Previous research found mixed aging results, depending on the metrics used to measure integration and segregation (Chan, Park, Savalia, Petersen, & Wig, 2014;Chen et al., 2021;Onoda & Yamaguchi, 2013;Zhang et al., 2021).Although most in the literature use the system segregation metric (Chan et al., 2014), no consensus exists surrounding integration.In general, the problem facing the integration-segregation framework is that there is no one way to define the two states.Many graph theoretical metrics could potentially be used (Rubinov & Sporns, 2010) and it is unclear why one should take precedence over the -2- other, particularly when their aging outcomes are mutually inconsistent.There is a need to more fundamentally define integration and segregation to transform it from a proxy to a physical quantity.
Here, we provide a physical foundation for the framework by applying the mean field Ising model to treat integration and segregation as physical 2-phase systems like magnets and liquids.After demonstrating that the Ising model can capture global brain dynamics as measured by functional MRI once the effective number of nodes is properly set, we proceed to calculate probabilities of being in the integrated or segregated states and find that younger and older brains are bounded by optimal and random signaling, respectively.We then explore diffusion and structural MRI data to ask if the age-related changes in signaling are due to changes in topological network connectivity.

APPLYING THE ISING MODEL TO FMRI
We model human brain signaling patterns obtained from resting-state functional MRI (fMRI) data sets.
As in previous work (Weistuch et al., 2021), we capture those patterns with the Ising model, a widely used theoretical method for expressing macroscale behaviors in terms of interactions among many underlying microscale agents (Dill & Bromberg, 2012).We first transform the continuous fMRI data into a representation as discrete Ising spins via binarization of the data (Figure 1).That is, we reduce the state of the region as either −1 or 1 based on whether fMRI signaling is decreasing or increasing, respectively.
Second, we calculate the synchrony by summing over all spins in a given time interval and dividing by the total number of spins (Figure 1).Synchronies are collected over the entirety of the scan to obtain a distribution.Based on Ising model theory, the synchrony threshold delineating between integrated and segregated states is set such that P int = P seg = 1/2 at the Ising model's critical point (Methods).P seg is the probability that the brain is in the segregated state and is defined as the relative number of time points for which the absolute value of synchrony is less than the synchrony threshold (Figure 1).P int is defined as the relative number of time points for which the absolute value of synchrony is greater than the synchrony threshold and trivially relates to P seg because P int + P seg = 1. -3-

RESULTS
The number of functionally effective brain regions Before proceeding to calculate P seg , we first check whether the model can capture the experimental synchrony distributions.A mean field Ising model only considering pairwise interactions has one quantity of interest.The strength of connection λ between any two regions corresponds to the degree to which signals between any two brain regions are correlated.However, we find that a naive fit of λ based on maximum entropy (Dill & Bromberg, 2012;Schneidman, Berry, Segev, & Bialek, 2006;Weistuch et al., 2021) fails to capture the synchrony distribution from fMRI data (Figure 2, orange).To improve upon a standard Ising model approach, here we introduce a hyper-parameter N eff .Brain atlas parcellations provide N brain regions, however, those N regions must be identically distributed across time for the Ising model to apply.We find that when setting N to a lower value N eff , fixed for all individuals within a data set, the Ising model accurately captures synchrony distributions (Figure 2).The optimal value of -4- N eff = 40 is determined by scanning across N eff multiples of 5 to find which best captures the next order moment not fit by our maximum entropy setup across all individuals (Methods, Figure 6).For our particular preprocessing (Methods), we find that N eff = 40 for individuals in the Cambridge Center for Ageing and Neuroscience (CamCAN) (Taylor et al., 2017) and the Human Connectome Project Aging (HCP) (Harms et al., 2018).For the UK Biobank (UKB) (Alfaro-Almagro et al., 2018), N eff = 30 performs best (Figure 6).
Based on identified N eff hyper-parameter values, brains act as if they have a few tens of functional units.If different preprocessing decisions are considered, such as atlas resolution, N eff values are still within an order of magnitude.At the voxel-level (N = 125, 879), we obtain an N eff value of 65 for CamCAN and 125 for HCP using the same procedure as for the Seitzman atlas (N = 300) considered in the previous paragraph (Figure S2).Future work will pinpoint how N eff depends on preprocessing to enable a future study creating a physics-based parcellation of the brain.
We also tried an alternative fitting strategy by fitting N eff per individual rather than having the same value for all individuals in a respective data set.We show that individually fitted N eff values trivially relate to λ as expected by theory (Figure S1).Moreover, individually fitted N eff are not found to be related to global differences in anatomical brain connectivity (Figure S3).The aging brain becomes functionally more segregated -5-  S1).Across the three publicly available data sets, we find that the balance shifts towards more segregation at older ages (Figure 3).Note that if we plotted P int rather than P seg , Figure 3 would be horizontally flipped, where P int goes from high to low values as a function of increasing age because P seg + P int = 1.
There is large variation among subjects (Figure S4).However, the correlation between age and P seg is significant with the largest coefficient being 0.40 for CamCAN, while the lowest being 0.08 for UKB.
Discrepancies in study designs may explain correlation magnitude differences: CamCAN and HCP are designed to study healthy aging (Bookheimer et al., 2019;Shafto et al., 2014), while the goal of UKB is to identify early biomarkers for brain diseases (Sudlow et al., 2015).
To better highlight how P seg changes across CamCAN's large age range, we present violin plots for younger, middle age and older individuals' P seg (Figure S5).We also investigate how P seg varies across -6-  S10).
At the critical point, we define P seg = 1/2 (Methods) and find experimental P seg values closer to 1/2 for younger individuals (Figure 3).Older individuals on the other hand, approach P seg = 1 on average.This limit corresponds to functionally uncoupled brain regions that are randomly activating.Our results support the critical brain hypothesis that healthy brains operate near a critical point (Beggs, 2022;Beggs & Plenz, 2003;Ponce-Alvarez, Kringelbach, & Deco, 2023;Tagliazucchi, Balenzuela, Fraiman, & Chialvo, 2012) and implicate aging as pushing brain dynamics further away from criticality.
Increasing segregation is not related to structural degradation In the previous subsection, we discussed the disruption of the integration and segregation balance from the perspective of phase transitions in physics.Here, we explore the physiological mechanism underlying increasing segregation in the aging brain.We consecutively simulate the Ising model on a hypothetically degrading brain structure and show that random removal of edges yields qualitatively similar results to those of fMRI (Figure 4).Note that Figure 4 is horizontally flipped from those of P seg (Figure 3) because average degree (relative number of edges) is on the x-axis.It is presumed that edges are lost as age increases.In Figure 4, edges are lost linearly in time, however, more complicated monotonic functions can be employed to yield a quantitative match with experimental data in Figure 3.We can also capture variability among individuals by assuming connection strengths within an individual are drawn from a distribution, rather than all being equal (Figure S11).In the supplement, we also demonstrate that similar qualitative trends are obtained when starting with other individuals' structures, regardless of their age (Figure S12).
-7- Orange data points on the right plot correspond to individual Ising systems, where N reflects the total number.The variable ρ corresponds to the Spearman correlation coefficient calculated over all orange data points between average degree and Pseg, with the p-value in parenthesis.Magenta data points correspond to medians, while error bars correspond to upper and lower quartiles for bin sizes of one degree.The schematic on the left is created with Biorender.com.
We now begin to investigate possible mechanisms of connection degradation.First, we find that our simulation is agnostic to the detailed mechanism of connection degeneration because connection strength is essentially modulated by the probability that a given edge exists (Figure S13).In other words, the simulation cannot inform whether connections are degraded based on some targeted property.Thus, we turn to structural MRI and diffusion MRI data from UKB to investigate possible properties being degraded with age.In Figure S15, we confirm that white matter volume decreases as a function of adult age, as previously reported (Bethlehem et al., 2022;Lawrence et al., 2021;Lebel et al., 2012).However, this decrease does not correspond to a loss of anatomical connections because we find that neither average degree, average tract length nor average tract density monotonically decrease with age when analyzing diffusion MRI scans using the Q-Ball method (Figure S16).This seems to contradict previous findings which report decreases (Betzel et al., 2014;Lim, Han, Uhlhaas, & Kaiser, 2015).However, -8- previous results employed the more simple diffusion tension imaging (DTI) method which is known to be less accurate at performing tractography (Garyfallidis et al., 2014;Jones, Knösche, & Turner, 2013;Rokem et al., 2015).When rerunning our analysis for DTI, we can reproduce previously reported tract properties' anticorrelations with age (Figure S16).We also investigate a graph property that captures polysynaptic connectivity called communicability (Andreotti et al., 2014;Estrada & Hatano, 2008;Seguin, Sporns, & Zalesky, 2023) and find that it also does not decrease age when using Q-Ball derived tract density (Figure S17).
We propose that observed white matter volume reduction (Figure S15) and brain dynamics change corresponds to less myelin covering axons as function of age.Despite rejecting anatomical connections as a possible mechanism in the previous paragraph, it remains inconclusive whether myelin underlies trends because we are not aware of such data being publicly available.Although axons are still physically present, myelin coverage disruption causes regions to no longer be functionally connected because signals do not arrive on time.Previously reported results from Myelin Water Imaging confirm reduction in myelin at advanced ages (Arshad, Stanley, & Raz, 2016;Buyanova & Arsalidou, 2021).We also investigated whether degraded functional connections are likely to be longer than average with age, as previously reported for certain brain regions (Tomasi & Volkow, 2012).Although we indeed find that the average correlation of the 25% longest connections is slightly more strongly anticorrelated with age compared to the average correlation of the 25% shortest connections for CamCAN (Figure S18, left), we find the opposite trend for HCP (Figure S18, right).Thus, myelin reduction does not seem to have a stronger impact on longer connections and conclude that the loss of functional connections happens randomly with respect to length at the brain-wide scale.

DISCUSSION
We apply the mean field Ising model to physically quantify integration and segregation at the emergent scale of the whole brain.From resting-state fMRI scans across three publicly available data sets, we find that brain dynamics steadily becomes more segregated with age.Physically, aging leads to brain dynamics moving further away from its optimal balance at the critical point.Physiologically, analyses of white matter properties point to random functional connection losses due to myelin degeneration as the possible culprit for more segregated dynamics.This expands upon our previous work finding metabolic -9-  2021).Future work will utilize N eff calculations to guide the creation of a parcellation in which brain regions are constrained to be physically independent based on their collective functional activity.
The field is inundated with integration and segregation metrics that have different aging trends.We go beyond heuristic definitions, such as one that we previously proposed based on matrix decomposition (Weistuch et al., 2021), by self-consistently defining the two states within the Ising model framework.
This makes our metric mechanistically based on the connection strength between regions and further stands out because P seg and P int are naturally at the emergent scale of the brain.We do not calculate a local property and then average over nodes to yield a brain-wide value ((Wang et al., 2021)'s metric also has this advantage).In addition, P seg and P int are directly related because P seg + P int = 1.Most integration and segregation metrics (Chan et al., 2014;Rubinov & Sporns, 2010;Tononi et al., 1994;Wang et al., 2021) are not defined to be anti-correlated.This could be advantageous because greater complexity can be captured (Sporns, 2010).
Taken together, it is not surprising that P seg and P int results are not consistent with some previous aging reports.For example, a property called system segregation, defined as the difference between inter-and intra-correlations among modules, was found to decrease with age (Chan et al., 2014).Although most report that segregation decreases with age, regardless of the specific metric (Chan et al., 2014;Damoiseaux, 2017;King et al., 2018;Zhang et al., 2021) (see (Chen et al., 2021) for an exception), integration trends are less clear.Global efficiency, taken from graph theory, was found to increase with age (Chan et al., 2014;Yao et al., 2019); however, others found different integration metrics decreasing -10- with results reported here.
The utility of the integration-segregation framework lies in its simplicity.However, its simplicity has led to various heuristic definitions that have qualitatively different aging trends.By physically defining integration and segregation based on connection strength between regions, we provide an interpretable foundation for more detailed studies going beyond the two-state approximation to investigate brain dynamics.

METHODS fMRI preprocessing
We access three publicly available resting-state functional MRI data sets: Cambridge Centre for Ageing and Neuroscience (CamCAN) (Taylor et al., 2017), UK Biobank (UKB) (Alfaro-Almagro et al., 2018), and Human Connectome Project (HCP) (Harms et al., 2018).Acquisition details such as field strength and repetition time can be found in Table S5.Demographic details can be found in Table S6.
UKB and HCP fMRI data are accessed in preprocessed form (for details see (Alfaro-Almagro et al., 2018) and (Glasser et al., 2018(Glasser et al., , 2013)), respectively).We preprocessed CamCAN data as done in our previous work (Weistuch et al., 2021).For all three data sets, the cleaned, voxel space time series are band-pass filtered to only include neuronal frequencies (0.01 to 0.1 Hz) and smoothed at a full width at half maximum of 5 mm.Finally, we parcellate into 300 regions of interest according to the Seitzman atlas (Seitzman et al., 2020).For our voxel-wide analysis presented in the Supporting Information, we do not perform parcellation and just consider gray mater voxels by masking.
Applying the Ising model requires the data to only take two possible values: −1 or 1.After performing the preprocessing outlined in the previous paragraph, we binarize the continuous signal for a given region based on the sign of the slope of subsequent time points (Weistuch et al., 2021).We previously showed that such binarization still yields similar functional connectivities as that of the continuous data (Weistuch et al., 2021).
Finally, we only consider brain scans that have the same number of measurements as the predominant number of individuals in the respective data set (Table S5).If the fitted connection strength parameter λ -11- is less than 0, reflecting a nonphysical value, we do not include that individual's brain scan in our analysis.In the HCP data set, we excluded individuals aged 90 years or older because their exact age, considered protected health information, is not available.
Identifying the N eff hyper-parameter In Figure 2  To identify N eff = 40 as the best value, we perform a parameter scan over multiples of 5 and identify the N eff at which the root mean square error (RMSE) between s 4 exp and s 4 model is minimized (Figure 6).We choose the fourth moment because it is the next order moment that our maximum entropy fit does not constrain.It is not the third moment because the distribution is assumed to be even as indicated by our prior (Equation S1).
-12- Calculating P seg The probability of the brain network being in the segregated state is the sum over all microstates corresponding to the segregated state.
In the second line, the mean field Ising model's P (n) is inserted (Equation S2).Z corresponds to the partition function and ensures that P (n) is normalized.The constant s * is the synchrony threshold for which segregated and integrated microstates are delineated.We set s * such that P seg = 1/2 when Λ = 0 according to theory.More specifically, we numerically calculate P seg (Λ = 0) for a given N eff and extrapolate to find s * (Figure S19).Proper calibration ensures that the theory is accurate and enables apples to apples P seg comparisons across data sets with different N eff .The list of s * values for the three publicly available data sets studied can be found in Table S1.The simulation for a given structure starts by randomly assigning the 64 nodes up or down spins.Then, for each time step, we attempt 10 spin flips 64 times, for a total of 2500 time steps.Spin flips are accepted according to the Metropolis-Hastings algorithm (Metropolis, Rosenbluth, Rosenbluth, Teller, & Teller, 1953).The exact number of spin flip attempts or total time points does not matter, as long as equilibrium is reached.For example, we find that for λ values larger than those presented in the text, synchrony distributions become asymmetric and exhibit only one of the two peaks corresponding to the integrated state because of the high kinetic barrier of going from all down spins to all up spins.
Although the starting structure is informed by diffusion MRI, resulting structures after computational edge removals are based on the posited removal strategy.Edges informed by dMRI are undirected and removal maintains undirectedness.Effectively two times as many edges are removed because both forward and backward edges are concurrently eliminated.In Figure S14, we demonstrate how synchrony distributions change as edges are computationally removed for a UK Biobank individual (subject ID: 6025360), with a starting λ = 34.4.

Diffusion MRI analysis
Diffusion MRI processing to obtain structural information such as tract length and streamline count, which we call tract density, is outlined in our previous work (Razban, Pachter, Dill, & Mujica-Parodi, 2023).Briefly, we take preprocessed dMRI scans from the UK Biobank (Sudlow et al., 2015) and calculate connectivity matrices using the Diffusion Imaging in Python software (Garyfallidis et al., 2014).We input the Talairach atlas (Lancaster et al., 2000) to distinguish between white and gray matter.

Ising model phase transitions
The Landau model is a general formulation to study phase transitions (Dill & Bromberg, 2012;Landau, 1937).It takes the following form, η corresponds to the order parameter.F is the free energy and can be expressed as the probability for being in microstate i by the following relationship F i = k b T ln P i .T is the temperature and T c corresponds to the critical temperature at which a second-order phase transition occurs.A and B are constants.
Here, we will express the Ising model's probability distribution (Equation S3) in terms of the Landau formalism (Equation S4) by approximating the binomial coefficient as an exponential to order (s 4 ).For brevity, we will write N to represent N eff .First, we use Stirling's approximation to expand the binomial coefficient.
To make further headway, we assume that s approaches 0 and expand Equation S9 to order s 4 . -2- Next, we assume N is large and express the term under the brackets as an exponential.
We can insert our approximate expression for the binomial coefficient back into P (s) (Equation S3) and obtain, Note that Equation S12 (after transforming into free energy space) maps onto Landau theory (Equation S4).s corresponds to the order parameter and λ c = N/2.At λ = λ c , P (s) switches from unimodal to bimodal, corresponding to a second order phase transition.We report a rescaled version of λ called Λ in Figure 5 and in other places in the Supporting Information to easily gauge how far an individual's connection strength is from the critical point.
Alternative N eff fitting approach Rather than choose one N eff for all individuals in the data set as done in the main text, we could fit N eff for each individual.Figure S1 demonstrates that such a procedure results in N eff values that are highly linearly related with λ.In other words, more precise N eff fits do not provide any more insight than maximum entropy fits of λ for all individuals in a data set under one optimal N eff .
-3- The N eff -λ relationship can be reasoned from the analytical expression for P (s) (Equation S12).When Λ < 0, which many individuals satisfy (Figure S10), P (s) is well-approximated as a Gaussian.
Thus, the analytical form for s 2 is: Since λ is fit in the Maximum Entropy framework to exactly match s 2 , Equation S15 indicates that a larger N eff requires a larger λ for a fixed s 2 .Indeed, we find in Figure S1 that the best fit line of the N eff -λ relationship has an approximate slope of 0.5, in agreement with Equation S15.-7-   2).Data points correspond to medians, while error bars correspond to standard errors for bins of 5 years.
The variable ρ corresponds to the Spearman correlation coefficient between age and Λ calculated over all N individuals, with the p-value in parenthesis.Segregation a network state limited to local signaling.
State a particular combination of physical properties.Here, we assume that brain networks can only occupy either the integrated or segregated state.
Ising model a classic model in physics that was first applied to ferromagnetism.It includes pairwise interactions between binary spin states.
Phase interchangeable with the word 'state' for the purposes of this text.
Critical Point the point where two phases coexist.In this text, it is where the synchrony distribution dramatically changes from bimodal (primarily integrated) to unimodal (primarily segregated).
Maximum Entropy fit a fitting strategy that satisfies user-defined constraints in the most agnostic way.
White Matter bundles of axons connecting brain regions. -1- signaling becomes less integrated and more segregated with age Authors: R.M. Razban, B.B. Antal, K.A. Dill, L.R. Mujica-Parodi Figure 1.Calculating the probability that the brain exhibits integrated or segregated dynamics (P int or Pseg).The schematic demonstrates the procedure for one individual's fictitious functional MRI scan with 4 brain regions and only two time points shown.First, we binarize data based on nearest

Figure
Figure created with Biorender.com.

Figure 2 .
Figure 2. Adjusting the number of brain regions (N eff ) helps capture experiment.The modified Ising model with N eff = 40 (yellow line) better captures the synchrony distribution (blue histogram) of an arbitrarily chosen individual in the Cambridge Centre for Ageing and Neuroscience data set (subject id: CC110045).The orange line corresponds to the Ising model with N equal to the number of regions in the Seitzman atlas (Seitzman et al., 2020).

Figure 3 .
Figure 3. Pseg rises in aging brains across three data sets.Data points correspond to medians, while error bars correspond to standard errors for bins of 5 years.The variable ρ corresponds to the Spearman correlation coefficient between age and Pseg calculated over all N individuals, with the p-value in Figure 4. Simulating the random removal of edges results in Pseg increases.Five edges are randomly removed from a starting diffusion MRI structure signaling becomes less integrated and more segregated with age Authors: R.M. Razban, B.B. Antal, K.A. Dill, L.R. Mujica-Parodi signaling becomes less integrated and more segregated with age Authors: R.M. Razban, B.B. Antal, K.A. Dill, L.R. Mujica-Parodi , our maximum entropy fit (orange line) fails to qualitatively capture the synchrony distribution for an arbitrary individual.To rescue the fit, we replace N with N eff (Equation S1).In the right plot of Figure5, we demonstrate that a mean field Ising model with N eff = 40 accurately captures the fourth moment of synchrony s 4 across all individuals in CamCAN preprocessed under the Seitzman atlas.Note that N eff is not a parameter like Λ; rather it is a hyper-parameter because it takes the same value across all individuals within the data set.N eff is necessary because the Ising model systematically underestimates s 4 when Λ > 0 (left plot of Figure5).Note that Λ corresponds to rescaling λ such that Λ = 0 is at the critical point (EquationS13).

Figure 5 .
Figure 5. Adjusting the effective number of brain regions (N eff ) helps capture synchrony distributions' variances across individuals in the Cambridge Centre for Ageing data set.Each data point corresponds to an individual.
Figure 6.The effective number of regions N eff is identified by minimizing the root mean square error (RMSE) of the fourth moment of synchrony between theory and experiment across all individuals.Each data point corresponds to the sum over all individuals' RMSEs in the respective data set.Note that the y-axis should be scaled by 10 −3 .
signaling becomes less integrated and more segregated with age Authors: R.M. Razban, B.B. Antal, K.A. Dill, L.R. Mujica-ParodiWe simulate the Ising model on an initial structure informed by diffusion MRI under the Harvard-Oxford atlas(Makris et al., 2006) (64 regions) for an arbitrarily chosen UK Biobank individual (subject ID: 6025360).If no edge exists between two regions, then the regions are uncoupled.If an edge does exist, then regions i and j are coupled and contribute λ * σ i * σ j to the system's energy; where λ corresponds to the connection strength and σ corresponds to the spin state of the corresponding region (−1 or 1).Under the standard notation of the Ising model, λ = J/T , where J corresponds to the coupling constant and T is the temperature of the bath.The starting λ is set to 34.4, which is above λ's critical point (starting P seg ≈ 0.2).By definition, N eff = N = 64 in the simulations.Based on atlas resolution, simulating the Harvard-Oxford atlas provides an N eff similar to those found for the experimental data (N eff = 40 for CamCAN and HCP; N eff = 30 for UKB).
signaling becomes less integrated and more segregated with age Authors: R.M. Razban, B.B. Antal, K.A. Dill, L.R. Mujica-Parodi signaling becomes less integrated and more segregated with age Authors: R.M. Razban, B.B. Antal, K.A. Dill, L.R. Mujica-Parodi signaling becomes less integrated and more segregated with age Authors: R.M. Razban, B.B. Antal, K.A. Dill, L.R. Mujica-Parodi signaling becomes less integrated and more segregated with age Authors: R.M. Razban, B.B. Antal, K.A. Dill, L.R. Mujica-Parodi Figure S1.Treating N eff as a parameter and fitting it per individual yields a strong correlation with λ.Each point reflects an individual brain scan and N reflects the total number analyzed.The value of the slope corresponds to that of the best-fit line for the data and is close to the predicted value of 0.5 (Equation S15).N is smaller than that of Figure 3 because some scans failed to have a minimum s 4 RMSE within the explored bounds of N eff (4-500) or λ values were nonphysical by being less than 0.

Figure S2 .
Figure S2.Identifying the effective number of regions N eff for brain scans processed at the voxel-level.Each data point corresponds to the sum over all individuals' RMSEs in the respective data set.

Figure S4 .
Figure S4.Pseg rises on average in aging brains but varies greatly among individuals with the same age.Blue data points correspond to individuals.The variable ρ corresponds to the Spearman correlation coefficient between age and Pseg calculated over all N individuals, with the p-value in parenthesis.Magenta

Figure S6 .
Figure S6.Standard deviations of Pseg per individual decreases as a function of age for CamCAN and HCP data sets.Data points correspond to medians, while error bars correspond to standard errors for bins of 5 years.The variable ρ corresponds to the Spearman correlation coefficient between age and Pseg calculated over all N individuals, with the p-value in parenthesis.Here, fMRI time-series data for an individual are equally split into 5 chunks and Pseg is

Figure S7 .Figure S8 .Figure S9 .
Figure S7.Pseg rises in aging brains across three data sets regardless of sex.Data points correspond to medians, while error bars correspond to standard errors for bins of 5 years.For UKB and HCP, we find that females' brains have higher shifted Pseg values across age.

Figure S11 .
Figure S11.Greater variance in simulations is seen when edges' connection strengths λ are drawn from a normal distribution with mean λ and standard deviation 3 * λ .At each consecutive step, λ is attenuated such that 5 edges are effectively removed per step ( λ = λ p edge ) from the same starting dMRI structure as in Figure 4 (UK Biobank subject ID: 6025360).Data points correspond to medians, while error bars correspond to standard errors for bins of 5 years.Orange data points on the right plot correspond to individual Ising systems, where N reflects the total number.The variable ρ corresponds to the Spearman correlation coefficient calculated over all orange data points between average degree and Pseg, with the p-value in parenthesis.Magenta data points

Figure S13 .
Figure S13.Edge removal mechanisms only matter in so much as they attenuate average degree for Ising simulations.In addition to randomly removing

Figure S16 .
Figure S16.White matter tract properties do not degrade as a function of age when using the Q-Ball method for tractography.However, they do degrade

Figure S18 .
Figure S18.For the Cambridge Centre for Ageing and Neuroscience data set, the shortest edges (lower quartile) have average Pearson correlations or average

Figure S19 .
FigureS19.The synchrony threshold s * is chosen such that it delineates between integrated and segregated states when Pseg = P int = 1/2 (red line) at the critical point (Λ = 0).This particular figure is created for 64 nodes; it must be set to the corresponding data set's N eff to determine the appropriate synchrony threshold.
Brain signaling becomes less integrated and more segregated with age Authors: R.M. Razban, B.B. Antal, K.A. Dill, L.R. Mujica-Parodi With an appropriately determined N eff , we can accurately set the same synchrony threshold s * for all individuals within a data set to calculate P seg .The value of s * is set such that at the Ising model's critical point in connection strength λ, P seg equals to 1/2 for the ideal synchrony distribution based on Ising model theory (Methods) .This enables P seg comparisons across data sets that may have different N eff values.For CamCAN and HCP, s * = 0.33 because N eff = 40 for both data sets.For UKB, s * = 0.36(Table We identify the best number of effective brain regions N eff such that the Ising model accurately captures individuals' synchrony distributions across the corresponding data set, improving upon our original application of the Ising model which lacked the N eff hyper-parameter(Weistuch et al., (Chong et al., 2019;Oschmann, Gawryluk, & Initiative, 2020; with age Authors: R.M. Razban, B.B. Antal, K.A. Dill, L.R. Mujica-Parodi with age(Chong et al., 2019;Oschmann, Gawryluk, & Initiative, 2020; Zhang et al., 2021), consistent . To generate the starting structure for Ising model simulations, we input the Harvard-Oxford atlas for tractography because it parcellates the brain into fewer regions, making it more computationally tractable to carry out simulations and closer to N eff values found for experimental data.Brain signaling becomes less integrated and more segregated with age Authors: R.M. Razban, B.B. Antal, K.A. Dill, L.R. Mujica-Parodi ageing and neuroscience (cam-can) data repository: Structural and functional mri, meg, and cognitive data from a cross-sectional adult lifespan sample.Neuroimage, 144, 262-269.

Table S1 .
Brain signaling becomes less integrated and more segregated with age Authors: R.M. Razban, B.B. Antal, K.A. Dill, L.R. Mujica-Parodi Data set values for Pseg calculations under our particular fMRI preprocessing procedure (Methods).

Table S2 .
Linear regression results for Pseg as a function of age

Table S3 .
Multiple linear regression results for Pseg as a function of age and sex across the data sets.

Table S4 .
Multiple linear regression results for Pseg as a function of age, sex and handedness for the Human Connectome Project.

Table S5 .
Functional MRI acquisition parameters of the data sets.

Table S6 .
Demographic information of the data sets for those individuals in Figure3.