Topological Data Analysis Captures Task-Driven fMRI Profiles in Individual Participants: A Classification Pipeline Based on Persistence

Abstract

BOLD-based fMRI is the most widely used method for studying brain function. The BOLD signal while valuable, is beset with unique vulnerabilities. The most notable of these is the modest signal to noise ratio, and the relatively low temporal and spatial resolution. However, the high dimensional complexity of the BOLD signal also presents unique opportunities for functional discovery. Topological Data Analyses (TDA), a branch of mathematics optimized to search for specific classes of structure within high dimensional data may provide particularly valuable applications. In this investigation, we acquired fMRI data in the anterior cingulate cortex (ACC) using a basic motor control paradigm. Then, for each participant and each of three task conditions, fMRI signals in the ACC were summarized using two methods: a) TDA based methods of persistent homology and persistence landscapes and b) non-TDA based methods using a standard vectorization scheme. Finally, using machine learning (with support vector classifiers), classification accuracy of TDA and non-TDA vectorized data was tested across participants. In each participant, TDA-based classification out-performed the non-TDA based counterpart, suggesting that our TDA analytic pipeline better characterized task- and condition-induced structure in fMRI data in the ACC. Our results emphasize the value of TDA in characterizing task- and condition-induced structure in regional fMRI signals. In addition to providing our analytical tools for other users to emulate, we also discuss the unique role that TDA-based methods can play in the study of individual differences in the structure of functional brain signals in the healthy and the clinical brain.

Introduction

Topological data analysis (TDA) provides quantifiable measures for capturing, understanding and analyzing the ‘shape’ of high dimensional data (Carlsson, 2009; Chazal & Michel, 2017; Edelsbrunner & Harer, 2008). TDA related methods are increasingly applied to characterize high dimensional biomedical data (Skaf & Laubenbacher, 2022) in domains as diverse as proteomics (Nowakowska & Kotulska, 2022), genomics and genetics (Nardini et al., 2021; Yu & Chang, 2021), and immunology (Sasaki et al., 2020). Recent reviews have strongly advocated for their extension to neuroscience in general (Sizemore et al., 2019) and for functional neuroimaging (fMRI) data in particular (Ellis et al., 2019; Salch et al., 2021). Applications to fMRI data are particularly valuable. As overt representations of hidden brain states (Friston et al., 2012; Tan et al., 2022), fMRI spatio-temporal signals are prototypical emergent properties of a general complex system (Von Bertalanffy, 1968). Understanding the properties of complex systems depends not simply on analyzing its constituents, but also on understanding how it’s constituents interact as a whole (Stephan, 2004). Therefore, TDA is a perfect tool for analyzing the structure and dynamics of network connectivity estimated from fMRI time series data (Anderson et al., 2018; Billings et al., 2021; Saggar et al., 2018, 2022), without the restrictions of assessing pairwise interactions between brain regions (Silverstein et al., 2016). Summarized measures of connectivity incorporate large numbers of features, and can therefore be used to characterize ongoing transitions in system states, or for inter-group classification (Centeno et al., 2022).

However, summarized network measures are not the only class of fMRI data that can be processed by TDA analyses. Rather, the dimensional structure (space, time and amplitude) of functional activity in individual brain regions is itself subject to task-induced modulation (Logothetis, 2008). Thus, the actual data itself (as opposed to processed data that provides information on network states) can be submitted to topological data analysis, and this forms the focus of our analyses. Here, we drove brain activity using a motor control task with distinct task conditions (Diwadkar et al., 2017; Meram et al., 2021; Morris et al., 2018). In two separate conditions, motor responses were elicited to visual probes that were presented in a) rhythmic (i.e., periodic or fixed) stimulus onset asynchrony (SOA) or b) pseudo-randomly varying SOAs. Using TDA, the functional profiles of the anterior cingulate (ACC, a region particularly salient in motor processing) in each task condition were summarized as persistence diagrams (PDs) (Cohen-Steiner et al., 2007). Following this, the PDs were transformed and summarized to persistence landscapes (Bubenik, 2015). These summaries were submitted for classification (using Support Vector Classifiers). In the remainder of the Introduction, we motivate our focus on the anterior cingulate cortex and provide a working summary of our TDA data processing pipeline. We also motivate the method’s value for reliably characterizing function structure in individual participants, thus complementing analyses based on group studies.

The Role of the Anterior Cingulate Cortex

Our investigation’s focus on the ACC was motivated by the structure’s close association with motor control, and the cognition – motor interface (Botvinick et al., 1999; Hoffstaedter et al., 2014; Paus, 2001). The cingulate cortex is characterized by highly variegated functions that vary across its ventral and dorsal aspects. The ventral ACC is most notably engaged by paradigms that evoke affective responses and demand affective control (Muzik & Diwadkar, 2016; Oliva et al., 2021; Tang et al., 2019), with the mid and dorsal ACC are most notably associated with motor and cognitive responding and control (Asemi et al., 2015; Hoffstaedter et al., 2014). Generally, the cingulate cortex’s anatomical positioning and its structural (and the resultant functional) connectivity allow it to effectively translate motivational and cognitive drives into motor plans and eventually into actions (Lam et al., 2022; Monosov et al., 2020; Paus, 2001).

The current paradigm was designed to specifically evoke differential motor response modes. Thus, probes presented with a periodic (i.e., fixed) SOA, result in low predictive complexity of evoked responses. This relative simplicity allows participants to implicitly develop a “motor set” (Diwadkar et al., 2017; Morris et al., 2018) that resonates to the stimulus cycle, and is presumed to drive “smooth” interactions between the cingulate and motor regions including the supplementary motor area (SMA) and primary motor cortex (M1) (Asemi et al., 2015). However, by randomly varying the SOA, predictive complexity is increased, breaking the “smoothness” of ACC (and network) engagement (Meram et al., 2021). Differences in the activation-based and network profiles of the cingulate during stimulus synchronization of uni-manual behavior have been extensively established from group analyses of fMRI data, implicating the structure in maintaining stimulus timing expectancy during motor control (Berchicci et al., 2020; Comstock & Balasubramaniam, 2018).

Processing fMRI Data

Established post-processing pipelines in fMRI can reliably discover a) task-driven activation profiles (Friston, 1995b; Goebel et al., 2006) and b) network repertoires modulated during task-active or task-free states (Silverstein et al., 2016). Activation-based analyses typically rest on the mass uni- or multivariate application of general linear models across the entire fMRI data matrix (Calhoun, 2001; Friston, 1995a, b). These analyses are designed to identify voxel activity changes, predicted by experimental conditions (tasks). However, because they are conducted voxel-wise, they are by nature agnostic about intrinsic topological structure in fMRI data (Salch et al., 2021). Moreover, approaches for group analyses do not provide a reliable framework for distinguishing between conditions at the individual participant level because such analyses are not optimized for identifying individual differences. Thus, after quantifying intra-subject effects at a first level, the statistical reliability of intra- or inter-group effects is addressed at a second level. While in recent extensions, individual activation profiles have been submitted for machine-learning based classification (Frankford et al., 2021), even here, first-level models are used to statistically filter data. By comparison, the sensitivity of persistent homology in detecting condition-driven differences in topological structure in individual participants can be achieved without statistical filtering or separating signal from statistical noise.

That topological structure is embedded within neuroimaging data was hinted at decades ago by Keith Worsley, in a paper that presaged the application of TDA to neuroimaging (Worsley, 1996). Worsley and others acknowledged that observable neuroimaging data consisted of a complex mixture of contextually driven spatio-temporal signals and must be characterized by intrinsic topology. Embedded within these signals lie hidden states of the underlying system (Friston, 2012). The functional properties of the signal play out in both space (across the brain) and time (across the duration of the fMRI acquisition) and are generally reflected in changes in signal amplitude. Therefore, as we and others have noted, persistent homology is ideal for capturing features that are intrinsic to the five-dimensional geometry of fMRI data (Abdallah et al., 2023; Ellis et al., 2019; Salch et al., 2021).

By applying TDA to five-dimensional data (three spatial dimensions in stereotactic space, one time dimension, and one dimension for signal amplitude), it is possible to characterize a) effects related to correlations between signal amplitudes across voxels in space, b) the dynamics of how those shapes change over time with changes in the experimental conditions of the task, and c) how these are related to changes in the fMRI signal.

An Analytical Toolkit Using TDA

Our specifically developed analytical toolkit uses a mix of topology and statistics to summarize regional fMRI signals (based on persistent homology, and persistence landscapes), before using machine learning (with support vector classifiers) to test for how well these methods summarize the data (Bubenik, 2020; Hensel et al., 2021). Notably, these tools are applied (and the results reported) for each individual participant. Thus, and as previously suggested, our toolkit a) motivates the search for individual differences in TDA-interrogated data structure thereby b) circumventing the search for group differences that inevitably depend on data summarization and a loss of individual features that is imposed by parametric statistics.

We rely on homology, a tool from algebraic topology, to make rigorous the notion of “holes” or “voids” in a shape (i.e., in the shape of the data). Persistent homology expands upon the notion of homology by tracking the changes to shape over a filtration of the signal, thus providing a multi-scale measure of signal shape (Carlsson, 2009; Edelsbrunner & Harer, 2008; Robinson, 2014; Wang et al., 2018). The changes to signal shape are recorded in a persistence diagram which encodes the scales at which emergent features of the signal appear and disappear. Persistence diagrams are then transformed into persistence landscapes (Bubenik, 2015), with the latter providing a computationally efficient framework compatible with commonly used statistical learning techniques. Two such techniques are permutation tests and support vector classifiers (Boser et al., 1992; Welch, 1990). These summaries allow for the classification of task conditions based solely on the geometry of the signal, as well as allowing for quantitative comparison with non-TDA based methods. This infusion of topological tools as an encoding of geometry may be viewed as a pre-processing step before statistical analyses. In contrast to other established preprocessing techniques however, our approach provides reliable identifiers between the geometric organization of the signal and the experimental conditions of the task. In a final step, we also compared the sensitivity of the employed TDA-based methods to non-topological based methods (that do not rely on persistent homology). As we will show, the TDA-approach out-performed the non-TDA approach in both the permutation tests and the support vector classifiers, suggesting that TDA is more sensitive to functional properties that are embedded in the shape of the data.

A comprehensive exposition of the toolkit is provided in the Methods section, and all code used in our analyses are publicly available. As we will argue in the Discussion, TDA provides a mathematically robust framework for capturing task-driven differences in the shape of regional and plausibly global fMRI signals and is a valuable complement to existing analytical toolboxes available to fMRI researchers.

Materials and Methods

Subjects

All study procedures were approved by the Institutional Review Board (IRB) at Wayne State University. TDA analyses were applied to data from eight randomly sampled participants, who were recruited as healthy controls for an ongoing clinical study (Ages: 17.2 – 19.4 yrs., Mean: 18.62 yrs.; 1 male, 7 females). Participants (or their Legal guardians) provided written informed consent prior to participating in the study. All participants were free of neuropsychiatric conditions, with absence of current or lifetime diagnosis confirmed using DSM-5 criteria (Association, 2013). By criteria, IQ in all participants exceeded the lower bound for mental retardation (≥ 80).

fMRI

Gradient echo EPI fMRI data acquisition was conducted at the Vaitkevicius Magnetic Resonance Center (WSU) on a 3 T Siemens Verio system using a 12-channel volume head coil (TR: 2.6 s, TE: 29 ms, FOV: 256 × 256mm², acquisition matrix: 128 × 128, 36 axial slices, voxel dimensions: 2 × 2 × 3 mm). In addition, a 3D T₁-weighted anatomical MRI image was acquired (TR: 2200 ms, TI: 778 ms, TE: 3 ms, flip-angle = 13°, FOV: 256 × 256mm², 256 axial slices of thickness = 1.0 mm, matrix = 256 × 256). A neuroradiologist reviewed all scans to rule out clinically significant abnormalities.

fMRI Data Acquisition

fMRI data were collected using an established uni-manual motor paradigm that employed a conventional block design (Amaro & Barker, 2006). During each task block (30 s), participants tapped their right forefinger to a flashing white probe using a receptive response box (Current Design Systems, Inc.). In Periodic epochs (fixed SOA) the inter-stimulus interval was fixed to the frequency of presentation (0.5 Hz: SOA = 2 s or 1 Hz: SOA = 1 s) allowing participants to establish and maintain a motor set. During Random epochs, the SOA was pseudo-randomly sampled from a Gaussian distribution (μ = 1.0/2.0 s; σ = 0.5/1.0 s), with lower bound set at 300 ms (to exceed the lower limit of response latencies typically noted in sequential finger movements) (Benecke et al., 1986). The random condition was designed to preempt a motor set and to elicit reactive responses under predictive uncertainty. The number of elicited responses was held constant across Periodic and Random epochs. Short Rest epochs (30 s) were interspersed between task-active ones.

fMRI Pre-processing

fMRI data were processed in SPM 12 using standard temporal (slice-time correction) and spatial preprocessing methods. For spatial pre-processing, EPI images were oriented to the AC-PC line, corrected for head movement through realignment to a reference image in the sequence, and co-registered to the anatomical high resolution T₁ image. Deformations from normalizing the high resolution T₁ image were applied to the co-registered EPI images to transform functional volumes to stereotactic space. A low pass filter (128 s) was used to account for low-frequency components. At the first level, epochs were modeled with boxcar stimulus functions convolved with a canonical hemodynamic response function to form regressors of interest. In each first level model, the six motion parameters (3 for translation and 3 for rotation) from the co-registration were modeled as covariates of no interest. Images were resliced (2 mm³) with a Gaussian filter (8 mm FWHM) applied. Images exceeding 4 mm of movement (< 1% of all images) were excised from analyses. Analyses were restricted to the anterior cingulate cortex using the established anatomical mask from the Automated Anatomical Labeling (AAL) atlas (Tzourio-Mazoyer et al., 2002).

TDA Background

A full appreciation of the current analyses is impossible without a brief overview of some key concepts underlying persistent homology (the Appendix contains a more thorough presentation of these ideas, and readers are directed to several survey articles in the literature (Carlsson, 2009; Chazal & Michel, 2017; Edelsbrunner & Harer, 2008; Ghrist, 2008; Munch, 2017)).

Cubical Complexes

A cubical complex is a geometric object built out of vertices, edges, rectangles, rectangular prisms, and so-forth. These building blocks are formally referred to as k-cubes, where k indicates the dimension of the object. Thus, a point is a 0-cube, a line segment is a 1-cube, a filled-in rectangle is a 2-cube, a solid prism is a 3-cube, and the higher k-cubes are defined analogously. A cubical complex is comprised of various k-cubes, as k varies (note that “cube” here does not imply that all the side lengths are equal).

For example, we can model the face of a chessboard as a cubical complex. The black and white squares serve as the 2-cubes, their edges are the 1-cubes, and their corners are the 0-cubes. This illustrates for instance, how geometric objects like the grid of a chessboard can be represented as cubical complexes. In this way, cubical complexes can be thought of as being similar to triangulations (except that the basic building block of a triangle is replaced by a rectangle).

As we glue k-cubes together in different ways, we construct different geometric objects. Even the simple case of gluing together two 0-cubes (points) and two 1-cubes (line segments) admits a variety of different geometries, which we now enumerate. Suppose first that the ends of both 1-cubes connected to different 0-cubes; the resulting space models a circle “decomposed” into two halves. On the other hand, if the two 1-cubes both connect to the same 0-cube, the resulting space is a “figure 8” together with a disjoint point. The disjoint point is represented by the lone 0-cube. Finally, if the ends of each 1-cube connect to a distinct 0-cube, the space models two disjoint circles. A figure 8, a circle with a disjoint point, and two disjoint circles have different topologies, reflecting the difference in their construction, even though they all started with the same “ingredients”.

The previously mentioned examples gracefully generalize to the voxel (i.e., volume element)—based structure of fMRI data. Here, each voxel serves as a 3-cube, each voxel face is a 2-cube, each voxel edge is a 1-cube and finally each voxel corner is a 0-cube. This representation allows an anatomical region of interest (such as the ACC) to be decomposed into a 3-dimensional cubical complex through the judicious choice of vertex, line segment, rectangle, and prism placement overlaid on the region. Here, we will model the ACC using cubical complexes, restricting ourselves to 3-dimensional cubical complexes (i.e., the maximum dimension of any present cube is 3). Figure 1 provides a graphical illustration of these concepts.

Fig. 1.

A schematic of the data transformation beginning from an fMRI acquisition and ending with a filtered cubical complex. a Activation is depicted for a single participant in the anterior cingulate (ACC). b Across time, a scalar value (signal amplitude) is attached to each voxel in the ROI. Each voxel can be viewed as a cube in a cubical complex, which is a decomposition of a region in combinatorial terms analogous to a triangulation. c The cubical complex is thresholded by signal amplitude. As the threshold changes, a family of cubical complexes emerges, nested within one another. Filtered cubical complexes form the input to persistent homology calculations

Filtered Cubical Complexes

In addition to its spatial properties (that permit the approach motivated above), each voxel also contains signal amplitude (represented as colors in the matrices in Fig. 1). In this way, fMRI data is 4-dimensional: one dimension for signal amplitude and three dimensions representing spatial coordinates (i.e., where in the brain the data is being collected). Signal amplitude is a generalized proxy for functional changes (either task-driven or spontaneous) in any brain region that contains any specific voxel. The importance of signal amplitude is exploited by filtering the region of interest (as described below). Filtering is crucial to our application because the signal amplitude represents an important dimension (specifically state changes) within the system. Thus, the simplified example in Fig. 1c, shows two (out of three) spatial dimensions with the dimension of signal amplitude represented in the color of each element.

A filtered cubical complex is a nested family of cubical complexes indexed on a real parameter s. In the present context, s denotes a threshold for signal amplitude and F_s is the set of voxels with amplitude greater than or equal to s. Using the natural cubical complex structure of each fMRI acquisition noted above, we have a cubical complex F_s for each integer signal amplitude s and for each time point. Agglomerating across signal amplitudes results in a filtered cubical complex {F_s} for each time point (an interactive visualization of a filtered cubical complex is provided in (Online Resource 2)).

We reasonably assume that different task conditions will induce differential functional structure in the fMRI data. Our goal is to study whether changes in the topological properties of {F_s} are sensitive to changes in task conditions. As we will show, thresholding the signal amplitude unpacks the topological properties of the highly complex data space {F_s} by crystallizing complex structures in the data. This crystallization will underpin functional inference.

To understand the nested nature of the filtered cubical complex {F_s}, let M denote the maximum signal amplitude and let m denote the minimum signal amplitude of a fixed time point. When s = M, the cubical complex F_M consists of only those voxels with signal amplitude equal to the maximum. At the other extreme, if s = m, then F_m consists of the entire cubical complex (i.e., the entire anatomical region under consideration), because every voxel’s signal amplitude is greater than or equal to the minimum, m. If we instead think of s as decreasing from M to m, then the cubical complex F_s grows, because more voxels appear in F_s as s decreases. Equivalently, more voxels in the underlying region satisfy the defining condition that their signal amplitude is greater than or equal to s. The power of persistent homology lies in the analysis of {F_s} which occurs between these two extremal bounds s = m and s = M. In the following paragraph, we highlight the importance of filtering with a hypothetical example.

We highlight our approach with two plausible empirical phenomena. In the first, there is a sub-region of voxels that is relatively higher in signal amplitude than its surrounding sub-regions. The isolated clump of high-amplitude voxels appears much earlier than the surrounding low-amplitude voxels as the threshold decreases from s = M to s = m. Hence {F_s} would contain this isolated clump and it would remain isolated for many values of s. For the second, there is a sub-region of relatively low signal amplitude surrounded by a sub-region with high amplitude. As before, the high amplitude voxels would appear earlier in {F_s} than the low-amplitude voxels. The filtered cubical complex {F_s} would contain a hollow, spherical shell type structure for many threshold parameters s. Only when the threshold is lowered would this hollow shell be filled-in. The topology of a hollow shell is different than a filled-in shell, and this difference can easily be detected using persistent homology. These two simplistic scenarios imperfectly begin to describe the relationship between the geometry of the signal and the topology of the resulting filtered cubical complexes {F_s}.

As previously hinted, filtering by signal amplitude allows the complex to grow or shrink based on the signal amplitude (i.e., the functional representation of system states). This allows us to isolate structures within the data and characterize the appearance and disappearance of these structures in a complex based on different thresholds. Thus, the geometric nature of the signal is reflected in the topology of the family of cubical complexes, and any observed geometric differences can be precisely quantified using persistent homology.

Homology

Homology is one of the primary tools used in algebraic topology to quantify the topological properties of geometric spaces. The output of homology is a sequence of positive integers known as Betti numbers and denoted b_i for i ≥ 0. The number b₀ counts the number of connected components and the number b₁ counts the number of ‘cycles’ (made precise later in this section). In our analyses, we restrict ourselves to the first two Betti numbers, but in general, one could study b_k for any k ≥ 0.

A connected component of a cubical complex consists of all k-cubes which can be connected to one another by a path completely contained within the cubical complex. The Betti number b₀ counts the number of disjoint ‘pieces’ of which a space is comprised. In Fig. 1b, the cubical complex consists of a single connected component, so b₀ = 1. However, in Fig. 1c (top), we see how two disconnected components merge into a single component (middle) following changes in the threshold. Thus, b₀ = 2 in Fig. 1c (top) but b₀ = 1 in Fig. 1c (middle).

A cycle in a cubical complex is a path along the 1-cubes that starts and ends at the same point. We say a cycle is trivial if it is the boundary of a collection of 2-cubes. For example, the boundary of a single 2-cube is a trivial cycle. In Fig. 1c (top), there is a single non-trivial cycle spanning the second, third, and fourth row of the cubical complex (the missing interior). By contrast in the same subfigure, the boundary of the isolated cube in the bottom left is filled in, thus the boundary is a trivial cycle. The number b₁ counts the number of nontrivial cycles in a topological space. Hence b₁ detects the number of “holes” in a cubical complex by counting cycles which surround those holes.

Persistent Homology

Persistent homology consists of the Betti numbers of F_s as a function of s. That is, for each time point, we compute the number of connected components and number of linearly independent loops of F_s. As s varies, connected components can appear or merge with other components. Similarly, loops can form out of connected components or become filled in as the threshold parameter s varies. The changes of geometry result in the appearance or disappearance of non-trivial cycles and/or connected components, and it is possible to track the threshold parameter s at which these changes occur. For example, we can determine the connected components which persist for large values of the threshold parameter, implying that these components may constitute sub-structures of functional interest.

The output of persistent homology is known as a persistence diagram, a collection of points in the Euclidean plane. A persistence diagram for a basic example is shown in Fig. 2a (left panel). The x-coordinate of each point is the threshold parameter when the feature first appears (is “born”) and the y-coordinate is the parameter when the feature disappears (“dies”). Points closer to the diagonal birth–death line y = x therefore correspond to features which do not persist through many threshold values. In contrast, points far from the diagonal line are long-lived features and therefore of potential functional interest. The persistence diagram provides a high-level topological summary of each time point, tracking when features are created and destroyed.

Fig. 2.

The figure depicts the pathway using which a persistence landscape is constructed from a persistence diagram. a In this schematic, starting from a simple persistence diagram consisting of only three points, horizontal and vertical lines are extended from each point in the diagram to the y = x (birth amplitude equals death amplitude) line. The graph is rotated 45° clockwise, so the y = x line is horizontal. This creates a set of right triangles whose hypotenuse lie on the new horizontal axis. The persistence landscape is the sequence of functions {λ_i(x)}, where λ_i(x) is the i^th maximum value of the set of triangles rotated graph at input x. Equivalently, λ_i is the i^th outer envelope. Both planar and three-dimensional versions of {λ_i} are plotted for clarity of visualization. b Observed data fed through the pipeline of (a) are depicted, highlighting the complex encoding the landscape provides of the initial data

Persistence Landscapes

A persistence diagram provides high-dimensional topological information in a succinct summary, but does not allow for basic statistical tests in any straightforward manner. Persistence landscapes offer one notable approach to addressing this limitation. Persistence landscapes provide a non-parametric vectorization scheme that can be efficiently computed and has easy-to-use implementations. Persistence landscapes are also stable with respect to noise, making them well-suited for studying noisy data (Bubenik, 2020).

Persistence landscapes associate a sequence of functions {λ_i}, i ≥ 1, to each persistence diagram. We step through their construction below (providing more details in the Appendix). To convert diagrams to landscapes, vertical and horizontal lines are drawn from each point in the persistence diagram to the y = x diagonal line (Fig. 2a, second from left). Next, the entire picture is rotated 45° clockwise, so the y = x diagonal line is now horizontal (Fig. 2a, third from left). The first, second, and so forth outer envelopes are constructed from this figure (blue, red and green lines in the landscape in Fig. 2a). We complete the process by setting λ_i to the i^th outer envelope function. The final panel displays a three-dimensional representation of the landscape function for ease of visualization. The process is partly reconstructed for experimental data in Fig. 2b.

Removing Common Features

The topological characteristics associated with spatial structure (i.e., the dimensions x, y and z of the anterior cingulate) are common across all participants (because data are normalized to stereotactic space), and these commonalities must be accounted for in any analyses. Genuine topological features are those points in the persistence diagram whose death coordinate is greater than the maximum signal amplitude M, and are thus seen as very tall peaks in the persistence landscape. To account for contributions of the fixed topological structure of the ACC, we removed those points in the persistence diagram whose death coordinate is greater than the maximum M in each degree of homology, before constructing the persistence landscapes and performing any statistical analysis. This correction allowed us to focus on the cumulative topological and geometric properties of the fMRI signal that include the 4^th dimension signal amplitude.

Statistical Testing

The last step of our analysis bifurcates into two statistical methods for classifying the persistence landscapes: 1) a labelled permutation test and 2) a support vector classifier. The goal of the labelled permutation test was to demonstrate that the canonical assignment of conditions (Periodic, Random, Rest) to computed landscapes was necessary to observe inter-condition differences. To demonstrate this, we performed a permutation test (1500 permutations) where the assignment of conditions to landscapes was randomly perturbed. Separately, a support vector classifier (SVC) was constructed based on the canonical assignment of conditions to the computed landscapes, allowing us to linearly separate this high dimensional data with a hyperplane (with classification accuracy with respect to this hyperplane assessed using 10-fold non-nested cross validation; full details of our statistical approach and hyper-parameter selection are provided in the Appendix) and scripts implementing our full workflow are available on GitHub.¹

Non-Topological Methods

To gauge the effectiveness (and in the case of individual participants, the “necessity”) of persistent homology for separately cross condition ACC profiles, we compared the TDA results by applying an SVC to data that were characterized using a non-TDA vectorization scheme (i.e., without persistent homology). These alternative characterizations were based on a consistent method for vectorization of the fMRI signal in each participant as follows: The dimensional coordinates (x, y, z) of every voxel within the ACC mask were used to derive a lexicographic ordering of the ACC, where each image in time provides a 7400-dimensional vector (based on the number of voxels in the region), with this collection of vectors used to train an SVC. The 7400-dimensional vector was constructed without TDA and was used for our baseline comparisons; we refer to this approach as non-TDA vectorized. Because SVCs are rotation-invariant with respect to their input data and the order of the voxels is consistent across patients, this choice of ordering does not impact our results. Note that the topological counterpart using persistence landscapes, was based on sampling the landscape functions at 1300 regular points spanning the entire range of their outputs (for comparison, the SVCs constructed using persistence landscapes lay in only a 1300-dimensional space as opposed to the 7400-dimensional space created by the ordering approach).

For reasons noted below, we were forced to omit the comparison of our TDA-based permutation test to a non-TDA analog of our permutation test. Our TDA-based permutation test relied on computing an average persistence landscape for each image in time, from which we could compute its norm and thus obtain a single number. Any other permutation test would also need to meaningfully associate a single number to each time point for valid comparison. Obviously, one cannot do this in any informative way using the fMRI signal by itself. Indeed, computing the mean, median, or any other simple statistic of an fMRI image would result in an extremely crude summary. We felt this was a vast oversimplification of the complicated nature of the signal and could not possibly provide meaningful validation of our methodology (or any other analytic pipeline), forcing omission of using a permutation test with our non-TDA based vectorization scheme.

Results

Permutation Test

Figure 3 displays that results of the permutation tests for each participant. The results are presented in heat maps of significance values (comparing differences between conditions) separately for each of the Betti numbers b₀ and b₁. In each map, inter-condition comparisons are represented in each column, with each row representing each of the eight participants. As seen, TDA was highly sensitive in distinguishing between ACC functional profiles for b₀, and in general, b₀ was more sensitive than b₁, a finding consistent with the literature (Edelsbrunner & Morozov, 2013). While each of the task conditions are reliably distinguished from the Rest condition, across participants, the test also reliably distinguished between the two task-active conditions. These results suggest that there is a high degree of reliability in the ability of b₀ and b₁ to distinguish between condition specific data at the level of individual participants.

Fig. 3.

The results of the permutation test for both Betti numbers b₀ and b₁ are depicted comparing the three experimental conditions. Darker red indicates better performance of the test. The data indicate that persistence landscapes successfully represented structure in the data induced by different task conditions and were sensitive in distinguishing between this structure across task conditions. As seen across all participants, the b₀ test expectedly performed slightly better than the b₁ test

Support Vector Classifier

Figure 4 displays the results from applying the SVC to a) non-TDA lexicographically vectorized and b) TDA summarized data. The heat maps here represent classification accuracy (ranging from 57 to 92%, with exact values reproduced in each cell). We implemented three-class classification (simultaneously separating between the three task conditions (also known as a one-versus-one classifier, first column), and pair-wise classification (between all pairs of the three conditions). Note that the accuracy of a random three-class classifier is 33%, and that of a random two-class classifier is 50%. Two inferences present themselves from the results: First, classification accuracy based on TDA-summarized data was high, particularly when distinguishing between pairs of conditions; and second, classification based on TDA-summarized data outperformed that based on our higher dimensional vectorization scheme for 88% of the cells (28 out of 32 comparisons). The second inference was further evaluated below.

Fig. 4.

The heat maps depict classification accuracies for both TDA and non-TDA vectorized based SVCs (darker red indicates greater accuracy). a The non-TDA vectorization SVC classification accuracies were generally poor (median accuracy = 62%). b The accuracies resulting from the TDA pipeline were high (median accuracy = 78%; Fig. 5 directly compares differences in classification accuracy). Notably, given the difficulty associated with multi-label classification, the multiclass, three-way classifier performed very well (median accuracy = 67%) and better than the non-TDA vectorized based classification (median accuracy = 49%)

Non-TDA Approaches

Figure 4 showed that TDA-summarized data greatly out-performed non-TDA vectorized data for most participants, and non-TDA vector classifiers were frequently indistinguishable from chance. Figure 5 and the accompanying analyses were constructed to evaluate statistically significant differences in classification accuracy between TDA- and non-TDA based classifiers. First we present comparative data for each of the eight participants (S₁ – S₈) within each of the four classifiers (Fig. 5a). As is evident from the 32 pairs of bars, TDA-based classification heavily out-performed non-TDA based classification, and differences across the 32 pairs of bars are depicted in Fig. 5b. Next, classification accuracy was summarized for each type of analyses within each classifier (Fig. 5c) and paired t-tests were used to distinguish accuracy for TDA- and non-TDA based classification. As seen, even with a small sample size, parametric analyses demonstrated that TDA-based classifiers significant outperformed their TDA-based counterpart for three of four classifiers. These results effectively summarize the benefits of including persistent homology (or alternatively the cost of omitting it) in the analyses of task-driven individualized activation data.

Fig. 5.

In quantifying the differences and gains that TDA provides, we show that TDA-based classification significantly outperformed its non-TDA vectorized counterpart. a The bar graphs compare the non-TDA (blue) and TDA (red) classification accuracies, grouped by test label for each of the eight subjects (S₁ – S₈). b From the classification accuracies in (a), we depict the relative difference for each participant, grouped by test label. As is evident, TDA significantly outperformed purely statistical approaches when comparing rest to non-rest modalities. c The bar graphs depict the means (± sem) data from the data in (a), for each of the four test labels. Even given the relatively small sample size (n = 8), we observed highly significant increases in the TDA-based classification accuracy for three of the four tests

Discussion

This multi-faceted investigation was motivated by multiple questions. These were: 1) Can persistent homology be used to summarize the topology of regional fMRI data? 2) Can these summaries be reliably classified such that different states evoked during task-driven processing are distinct? 3) During classification of regional profiles, does TDA summarization out-perform non-TDA vectorizations and 4) Can these methods be robustly applied at the level of individual participants, thereby providing a sound mathematical framework for future studies of individual differences of clinical pathologies? The answers suggested by our results are reassuring: 1) The pipeline described in Figs. 1 and 2 (and the Appendix) successfully employed a sequential array of topological methods, where changes to the shape of fMRI signals were persistence diagrams, before the latter were transformed into persistence landscapes for classification; 2) Subsequent classification of profiles of the ACC during a motor control task was highly reliable in distinguishing between three unique task conditions, and 3) outperformed classification when data from the ACC were summarized using a lexicographic vectorization scheme. 4) TDA-based classification provided a mathematically robust framework for distinguishing between regional profiles of the ACC for most of the individual participants (see Figs. 3, 4, and 5), motivating possible future applications in healthy and clinical samples. In the remainder of the Discussion, we revisit some of the theoretical motivations for our approach. We focus more on discussing the mathematical and computational underpinnings of our study, rather than its functional interpretations, noting that the role of the anterior cingulate cortex in motor processing has heavily discussed elsewhere (Diwadkar et al., 2017; Gonzalez & Burke, 2018; Muller & Anokhin, 2012; Smith, 2021).

Topological Nuances

In the current application, topological analyses proceeded after filtering the cubical complex (formed from the ACC) by signal amplitude. Notably, the signal threshold is not fixed, but varies continuously through all possible choices of threshold (see Fig. 1). Continuous variations in the threshold create sub-regions in any cubical complex, permitting analyses of their topological properties. Specifically, this study focused on the number of such sub-regions (b₀) and the number of loops (b₁) as parameterized by a signal threshold s. By encoding b₀ and b₁ in a persistence diagram, they can subsequently be transformed into persistence landscapes (Fig. 2) which are stable and computable vectorization of diagrams, and are well-suited for signal analysis (Robinson, 2014).

The ability to describe the ACC (or indeed any region of the brain) as a cubical complex is essential to our approach. This specific choice of discretization provides a clear, one-to-one mapping between a region of study (e.g., the ACC) and a topological abstraction of it. More generally, by respecting the intrinsic dimensions of fMRI data (with three spatial dimensions, a temporal dimension, and a dimension of signal amplitude), TDA-based analyses provide a complementary framework to that involving the detection of signal changes in fixed spatial coordinates. Rather, by including all five dimensions as part of the analyses and by filtering the data space by signal, our efforts attempt to discover how location and time drive changes to signal structure. Thus, in using persistent homology the process of discovery emerges from the cumulative (i.e., five-dimensional) shape of the acquired data. It is not that TDA explicitly “competes” with established techniques for functional discovery of regional brain activity. Rather, as noted TDA provides a fundamentally different framework for a) conceptualizing data (e.g., filtered cubical complexes) before b) providing the formal mathematical tools (e.g., persistent homology and persistence landscapes) for discovering and quantifying structure within the data.

If the functional profile of fMRI data is intrinsically high dimensional, and if our pipeline is inherently sensitive to this dimensionality, then it is unsurprising that TDA-based classification outperforms the alternative summarization scheme. As was seen, across participants, topologically informed SVCs generally out-performed their non-topological counterparts, in both bi-modal and three-way classification schemes. The continuously varying threshold allows changes in signal to be identified within the region without fixing a single threshold parameter. Further, these changes present themselves as persistence landscapes with very tall peaks and wide bases that persist across many signal threshold values. Persistent homology doesn’t rely on any assumptions about data being linearly distributed (Edelsbrunner & Morozov, 2013) and characterizes structure intrinsic to the signal itself. This can be contrasted to many traditional statistical and vector-based approaches (Boser et al., 1992; James et al., 2013).

A final notable aspect of our workflow is its ability to promote meaningful inferences from fMRI data in individual participants. Characterization of fMRI signals in individual participants is both a conceptual and methodological challenge. Conceptually, frequentist statistics are not naturally suited for studying fMRI signals, because inference from fMRI data must rely on models of how the observable data are “caused” (Stephan et al., 2006; Woolrich, 2012). Moreover, fMRI signals are subject to variations in the hemodynamics of signals across the brain. This variability drives different activation characteristics across the brain that are unrelated to functional brain organization, even within participants (Bielczyk et al., 2017; Lin et al., 2018). Among the many downstream effects of these problems are the inflation of false positive rates in cluster based inference (primarily for resting state studies) (Eklund et al., 2016) and the proposed need to rely on very large sample sizes for reproducibility of effects (Marek et al., 2022). Methodologically, many compelling approaches to characterizing individual differences have relied on connectomics rather than activity in individual regions (Finn & Rosenberg, 2021). Our approach does not obviate these concerns, but partly circumvents them. First, our workflow includes all the acquired data independent of threshold and is therefore not subject to statistical thresholding at the point of analyses. Second, we suggest that we can be agnostic about generative models of the acquired fMRI signals. This is because our analyses are not motivated to discover functional mechanisms (Mannino & Bressler, 2015), but are designed to achieve a maximally sensitive summarization of task-driven system states. Our results suggest that the pipeline achieves this (at least for the current data set).

Limitations and Future Directions

fMRI is poised to remain the imaging modality of choice at least in the fields of psychology, psychiatry and neurology. With the expansion of imaging consortia, there is greater access to large data sets, with a greater possibility for probing meaningful individual differences (Bethlehem et al., 2022; Essen et al., 2012). Most of these approaches focus on resting-state functional connectivity (Bruin et al., 2023), yet the study of task-driven differences in regional brain profiles remains highly valuable because tasks drive the brain and therefore, drive fMRI signals (Logothetis, 2008). Thus, regional brain responses represent the relative specialization of brain function (Friston, 2005), and from this relative specialization and patterns of connectivity in the brain, emerge functional network interactions (Schotten & Forkel, 2022). Moreover, studying regional activity can assume importance either in disorders of focal brain function, or in studying regional responses in brain structures that are strongly associated with clinically important dimensions (e.g., the amygdala in mood or depression, or the insula in interoception) (Craig, 2011; Nord et al., 2021; Soloff et al., 2017).

TDA is limited because it is an “agnostic” tool for functional, but not mechanistic discovery. The analytics are designed to extract information on the structure of data, but are agnostic about how the data were “caused”. Accordingly, in our analyses, BOLD data were accepted as having arrived through the filter of hemodynamics, yet containing within its structure, credible information about functional system states. Thus, TDA does not offer the possibility to learn something unique about brain mechanisms per se but is limited to summarizing the structure of data. Yet, these limitations are somewhat outweighed by our demonstrated sensitivity to task conditions and the generalizability of these sensitivity across individual subjects. A general aim of our work in TDA is to motivate its application to the analyses of regional fMRI signals (Abdallah et al., 2023; Salch et al., 2021). In this extension, we have harnessed TDA to derive individualized regional profiles of the anterior cingulate cortex in task-driven states, while showing that these profiles are reliably classifiable using support vector machines. We hope that the report and the publicly available code will motivate readers to use TDA in their own studies of individualized regional brain function in health and disease.

Information Sharing Statement

As required by National Institutes of Health, all fMRI data collected for the current analyses have been deposited in the NIH Data Repository (MH059299). In addition, all code used to analyze the data and create all of the pipelines discussed is made publicly available at https://github.com/catanzaromj/PL_fMRI.

Appendix

In this appendix, we describe the mathematical details of our methodology and place persistent homology on a formal footing. Persistent homology calculations are frequently performed with binary arithmetic to increase the speed of calculations and we do the same throughout. In particular, our vector spaces have binary coefficients; so, 1 + 1 = 0.

Homology.

Let X denote a cubical complex, e.g., a single stage F_s of a filtered cubical complex as described in the main text. The vector space of k-chains of X, denoted C_k(X), is the vector space with basis given by the set of k-cubes. Thus, C₀(X) is the vector space with basis given by vertices in X and C₁(X) is the vector space spanned by the edges of X. To connect these adjacent degrees, we use an algebraic notion of boundary inspired by the geometric version. The boundary of a k-chain is the sum of the (k-1)-dimensional faces of each of its k-cubes. By definition, we set the boundary of a 0-chain to be 0. For example, the boundary of a 1-chain is given by the sum of endpoints of each 1-cube within the 1-chain. Hence, the boundary maps links together k and (k-1)-chains for all k.

We say a k-chain is a k-cycle if its boundary is 0. Note that a sequence of edges forming a geometric cycle is a 1-cycle because each vertex is a face of two (consecutive) edges and in our binary arithmetic, 1 + 1 = 0. The set of k-cycles is denoted Z_k(X). We say a k-chain is a k-boundary if it is the boundary of a (k + 1)-chain. The set of (k + 1)-boundaries is denoted B_k (X). A fundamental observation of algebraic topology is that every boundary is also a cycle. That is, the boundary map applied to a k-boundary is zero. Thus, the vector space of cycles is a sub-vector space of the boundaries and we can form the vector space quotient

$${\text{H}}_{\text{k}}\left(\text{X}\right)={\text{Z}}_{\text{k}}\left(\text{X}\right)/{\text{B}}_{\text{k}}\left(\text{X}\right).$$

The quotient vector space H_k(X) is known as the cubical homology of X in degree k. We provide examples of these definitions in the following paragraph.

For example, a 0-dimensional cycle can simply be a point or vertex. A generic 0-dimensional cycle is a formal algebraic sum of such vertices. There are no 0-boundaries. If two vertices are the endpoints of a line segment, then the sum of those two vertices is a 0-boundary, and those two vertices correspond to the same element in the quotient H₀(X). Generically, vertices may be connected by consecutive 1-cubes rather than a single one. Two vertices are homologous or equivalent if there is a path in the cubical complex connecting them. The set of all vertices homologous to one another form a homology class. Every vertex lies in exactly one homology class. Elements of 0-dimensional homology are not vertices themselves, but rather collections of vertices which are homologous to one another. If any two points in the complex can be connected by a path, then every point is homologous to every other, and there is only one such collection of vertices. In this case, the zeroth Betti number b₀ = 1, reflecting the fact that there is just one connected component. Alternatively, if the cubical complex consists of N disjoint pieces, then b₀ = N. Thus, 0-dimensional homology measures the intuitive notion of being “connected”, meaning the complex consists of a single piece, and the Betti number b₀ counts the number of components or disjoint pieces.

Stepping up one dimension, we have 1-dimensional cycles, 1-dimensional boundaries, and their quotient 1-dimensional homology. Examples of 1-dimensional cycles include paths of 1-cubes which start and end at the same 0-cube and formal linear combinations of such paths. Two 1-dimensional cycles are homologous if their sum is a 1-boundary. If a 1-cycle surrounds a filled in region, so that it is the boundary of that region, then it is a 1-boundary and thus trivial in H₁. It is said to be trivial. The Betti number b₁ counts the number of non-trivial equivalence classes of 1-cycles.

In Fig. 6, two 1-cycles are highlighted in gray and blue. The grey cycle encloses only voxels which are active within the region. The cycle is therefore a boundary of these voxels and thus the gray loop is trivial. Since the cubical complex does not contain the omitted (white) squares, the blue cycle is fundamentally different from the gray—it is non-trivial, since it cannot be expressed as the boundary of a collection of 2-cubes. Any attempt to express it as such a 2-boundary would require the “missing” voxels. The blue cycle is equivalent to many other loops including the boundary of the two omitted squares. We see this by noting the difference between the original blue cycle and the boundary of the omitted square is precisely the set of 2-cubes enclosed between the two cycles. These two cycles are equivalent from a homological perspective, and persistent homology counts them as one in the same, since they both enclose the same topological hole. Thus b₁ = 1 for the region displayed in Fig. 6 since there is a single non-trivial cycle. For completeness, b₀ = 1, since the region is connected.

Fig. 6.

The figure displays a filtered cubical complex. The two-dimensional cells (square voxels in this case) are colored by their signal amplitude, with darker red corresponding to higher signal amplitude, and yellow corresponding to lower signal amplitude. White regions correspond to voxels whose signal amplitude does not satisfy the threshold constraint and thus are not within the region. Two loops are highlighted. The grey loop on the left bounds voxels completely contained within the region so it is a trivial loop and does not contribute to b₁. In contrast, the blue loop bounds voxels which are not within the region (the excluded white voxels) and contributes to b₁

An Example.

We illustrate our topological constructions with an example. A two-dimensional filtered cubical complex is displayed in the interactive display of (Online Resource 1), where the color of a square indicates its signal amplitude (filtration) value. (Online Resource 1) provides a slider below the figure which allows the user to vary the threshold interactively and see the resulting cubical complex. We compute the persistence diagram for b₀ and b₁ of this small example. The initial cubical complex, with threshold parameter B_max = 10, has four connected components with no non-trivial loops. Hence, b₀ = 4 and b₁ = 0. Decreasing the signal amplitude threshold to 9 introduces three additional squares which in turn changes to the resulting topology. Two of the four components from the previous threshold are now merged. Furthermore, a loop has now formed in the bottom right. Thus, b₀ = 3 and b₁ = 1 for a signal threshold of 9. As we continue to decrease the signal threshold, the topology of the cubical complex continues to change. At a signal threshold of 5, the cubical complex becomes connected, so b₀ = 1. In particular, the loop formed at a threshold of 9 lives until the threshold becomes the minimum B_min = 1. The loop surrounding this square lives for a relatively long duration. This is indicative of a square (or more generally a region) of low activation being surrounded by an area of high activation. Such features are highlighted by persistent homology and appear in the persistence diagram as points very far from the y = x line.

Our topological analysis of fMRI signals follows the same flow as in the example of (Online Resource 1). We begin with a maximum threshold determined by the largest signal amplitude. As we reduce the signal amplitude threshold, we track how the topology of the resulting cubical complex changes. An interactive visualization for fMRI signal in the ACC is shown in (Online Resource 2). Starting from the maximum threshold B_max = 1900, there are three connected components. As the signal threshold decreases, new connected components are formed and merged together. The persistence diagram summarizes these changes which are then reflected in the persistence landscape.

Persistence Landscapes.

The persistence diagram provides a useful visualization of the geometric properties of a filtered cubical complex. However, standard statistical analysis applied to persistence diagrams is not always straightforward or interpretable. To remedy this, vectorization schemes were developed, allowing persistence diagrams to be embedded into vector spaces where standard statistical tests can be applied. One such vectorization scheme is known as the Persistence Landscape. Persistence landscapes provide a convenient, non-parametric vectorization of the diagram with no additional parameters to tune. They are essentially a repackaging of the same information provided by persistent homology amenable to arithmetic manipulation. Importantly, they provide a stable avenue to apply the techniques of machine and statistical learning to the output of persistent homology.

A persistence landscape constructs a sequence of piecewise-linear functions λ_k from a given diagram. Explicitly, for each point (b,d) in the diagram, first define the auxiliary function.

The graph of each f_(b,d) function is an isosceles triangle whose base is precisely the interval (b, d) and whose height is determined by d-b. Finally, the persistence landscape is the sequence of functions {λ_k}, where each λ_k is a real-valued function defined so λ_k(t) is the k^th-largest value of the collection of auxiliary functions{f_(b,d)(t)}. Therefore, λ₁(t) is the largest value of {f_(b,d)(t)} for each t, λ₂(t) is the second largest value of {f_(b,d)(t)}, and so on.

As functions can be added, subtracted, and multiplied by real scalars, they naturally exhibit a vector space structure. Moreover, it is easy to construct the mean persistence landscape of a given set of persistence landscapes (whereas a mean persistence diagram might fail to exist). This allows us to easily perform statistical tests (e.g., permutation tests) on a set of labeled persistence landscapes which would be much more difficult on the underlying diagrams.

The conversion of a persistence diagram to a persistence landscape just described is shown in Fig. 2. Starting in Fig. 2a left, we see a simple persistence diagram. The vectorization procedure begins in Fig. 2a second from left, by drawing horizontal and vertical lines from each point in the diagram to the diagonal line y = x. The entire picture is rotated in Fig. 2a middle. We take outer envelopes of the dashed piecewise-linear lines of Fig. 2a middle to construct a sequence of functions λ_k, k ≥ 1. Thus, λ₁ consists of the outer-most envelope, λ₂ consists of the second outer-most envelope, and so forth (Fig. 2a right). The entire sequence of functions λ_k is the persistence landscape. In practice, each of these functions is sampled at a large number (roughly 1000) of points and concatenated to construct a single vector. We use the persim package of scikit-tda to compute the persistence landscape in practice. The procedure is the same for actual experimental data, albeit the output is more complicated (Fig. 2b).

Statistical Testing

We now describe in detail our labelled permutation test and support vector classifier (SVC) which comprise the main statistical tests. In addition, we also performed LASSO tests on the data.

Permutation Tests.

We perform a total of six labelled permutation tests, each specified by a pair of task conditions from (Periodic, Random, Rest) and a homological degree (0 or 1). The first step in each test is to construct the baseline significance. This is done by computing the average landscape of the selected conditions individually, remembering that landscapes can be treated as vectors in a vector space where arithmetic operations are well-defined. The baseline significance is the supremum norm of the difference of the averages. The second step is to perform a stratified shuffle of the labeling on the original landscapes, so that the correct ratio is maintained between the labels. For each shuffle, the above procedure is repeated: compute the average landscape with respect to the shuffled labelling and compute the supremum norm of their difference. If the supremum norm from a shuffled labelling is greater than the baseline significance, we deem the shuffled labelling to be significant. After 1500 shuffles, the p-value of the test if the number of significant shuffles divided by 1500. These p-values are reported in Fig. 3.

Support Vector Classifiers.

The support vector classifiers also depend on a pair of task conditions; however, the landscapes are concatenated along their homological degree for this test. In particular, if λ⁰ and λ¹ are the landscape vectors in degree 0 and 1 respectively, then the vector λ^tot = (λ⁰, λ¹) is used for construction of the SVC. Note that since λ^tot contains both λ⁰ and λ¹ there is no loss in topological information in doing the concatenation. Each SVC is validated using tenfold cross-validation, a commonly used technique in machine learning. For each pair of test conditions, only 90% of the labelled data is used in the construction of the SVC. The remaining 10% is used to test the validity of the SVC in terms of its classification accuracy—we count how many of the remaining 10% of the data was correctly classified. This process is repeated 10 times or folds, so that at the end, the SVC algorithm has been tested against every data point exactly once. The classification accuracy from each of the 10 folds are averaged, and the results are displayed in Figs. 4 and 5.

The hyper-parameters of each SVC were determined by performing a standard grid search with 5-fold cross validation over the regularization parameter (typically denoted ‘C’) and the kernel type (whether ‘linear’, polynomial ‘poly’, or ‘sigmoid’). For the polynomial kernel, degrees less than 4 were also sampled over.

LASSO.

We also tested our topological approach through the lens of a LASSO classifier. LASSO (least absolute shrinkage and selection operator) regression is a statistical analysis method that allows for variable selection by enforcing an L₁ regularization penalty in its optimization routine. This leads to enhanced interpretability of the results and improved summary of the data. In particular, LASSO provides in-built feature selection, useful for analyzing high-dimensional data. Figure 7 displays the classification accuracies of a LASSO classifier applied to our persistence landscape (TDA-based) summary versus our non-TDA vectorized approach. The TDA-based approach outperforms the non-TDA vectorization in almost every case. More generally however, LASSO underperforms the SVC pipeline (TDA and Vectorization, see Fig. 4). This is unsurprising because a LASSO classifier constructs a simple linear classifier which is a simpler statistical model than an SVC (with non-linear kernel).

Fig. 7.

The heat maps for LASSO classification accuracies for TDA and non-TDA vectorized approaches (darker red indicates greater accuracy). a The non-TDA vectorization LASSO classification accuracies were above average (median accuracy = 59%). b The LASSO accuracies resulting from the TDA pipeline were higher (median accuracy = 62%) but still underperformed our SVC results. There were no discernible trends in comparing the multiclass, three-way classifier in either case. Comparing with Fig. 4, we find SVCs provide a stronger predictor than LASSO

Data Availability

The datasets generated during and/or analyzed during the current study are available from the corresponding author on reasonable request.