This article proposes a formalization of psychological functions as Topological Functional Systems, drawing on the historical–cultural tradition and functional systems theory, and integrating them with tools from contemporary topological dynamics. First, it reviews the theoretical foundations that conceive psychological activity as a systemic, dynamic organization oriented toward adaptive outcomes. It then develops a mathematical formalization that accounts for functional invariance under variable modes of execution and network reorganization. On this basis, the article derives clinical implications for neuropsychological assessment and rehabilitation, emphasizing functional reorganization rather than localized deficit. Finally, it discusses the scope and limitations of the model in dialogue with current approaches in neuroscience and neuropsychology.
Contemporary neuroscience is organized, in a cross-cutting manner, around two fundamental organizational principles: the principle of functional segregation, which refers to the relative specialization of brain regions and circuits in specific processing domains, and the principle of integration, which denotes the dynamic and context-dependent coordination of multiple distributed areas for the production of coherent functions at the system level (
A FS is defined as a dynamic organization of the nervous system that is constituted and regulated according to the attainment of a useful adaptive outcome, which acts as the central organizing principle of the system (
This property—the variability of the components that sustain a function while preserving the function itself and its adaptive goal—becomes particularly evident in both development and pathology following brain injury. At the developmental level, the neuroscience of language has shown that the networks supporting linguistic functions do not exhibit a stable organization from early childhood, but instead undergo progressive transformations as the nervous system matures, in a manner consistent with the Vygotskian idea that higher functions emerge through the integration and reorganization of earlier structures (
This property endows FS with a topological structure, insofar as their identity is not defined by the fixation of components or by specific configurations, but by the preservation of invariant functional relations that allow the same adaptive useful result to be achieved through variable realizations. This conception is implicit in Pyotr Anokhin’s theory of FS, in which the result dynamically organizes and selects the components of the system, and it is expressed with particular clarity in Nikolai Bernstein’s theory of movement, which showed that a single action can remain functionally stable despite variability in its execution patterns. In both cases, what remains is not the fixed composition of the system, but a set of stable functional properties whose preservation confers continuity and identity to the system beyond its particular realizations.
In continuity with this conception, the aim of the present work is to provide a foundation for a formalization of FS as Topological Functional Systems (TFS), in which the functional identity of psychological processes is expressed in terms of dynamic–topological invariants rather than fixed anatomical configurations or local mechanisms. This formalization seeks to rigorously articulate the neuropsychological tradition of functional systems with tools from topological dynamical systems theory, so that variability in components and modes of execution can be understood as a structural property of the system, rather than as a deviation or a failure of control. At the same time, this perspective aims to offer a conceptual framework with direct clinical implications, in which phenomena of development, functional reorganization, and pathology due to brain injury can be interpreted as dynamic transformations of the system that preserve—or alter—its functional invariants. From this standpoint, neuropsychological symptoms are not reduced to deficits localized in isolated components, but are conceptualized as expressions of changes in the trajectories and functional configurations of the system as a whole, thereby opening new avenues for clinical understanding and intervention grounded in systemic, dynamical, and topological principles.
The article is structured into three main sections. The first, “Background and Theoretical Foundations,” presents the foundations of FS and their articulation with the systemic, social, and dynamic conception of psychological functions developed by Vygotski and formalized in the work of Alexander Luria. The second section, “Mathematical Foundations,” introduces a formalization of FS as TFS, articulating the architecture and functionality of FS with tools from topological dynamical systems theory. Finally, the section “Clinical Implications” discusses the scope of this formalization for the neuropsychological understanding of symptoms, proposing a clinical reading centered on the variability of modes of execution and the preservation of functional invariants, with implications for assessment and intervention from a systemic perspective on psychological activity.
In the field of neuropsychology, the present approach seeks to provide continuity and contemporary projection to the scientific program of the historical–cultural tradition, recovering its fundamental orientation according to which psychological and neural activity is constituted through processes of social origin, organized systemically, and dynamically regulated in relation to adaptive outcomes. From this perspective, the proposed Systemic Neuropsychology places TFS at the center of its inquiry as fundamental theoretical units for understanding the organization and variability of psychological functions. Within TFS, the classical foundations of Anokhin’s theory of FS, Vygotski’s historical-cultural conception of psychological processes, and their neuropsychological elaboration in the work of Luria are articulated and integrated with contemporary contributions from neuroscience, dynamical systems theory, and topology. In this sense, the aim of the present work is to establish the theoretical foundations and the mathematical assumptions required to formalize FS as TFS, thereby offering a conceptual framework capable of sustaining clinical and neuropsychological interpretation consistent with the dynamic, variable, and systemic nature of psychological activity.
The systemic conception of the psyche elaborated by Vygotski and Luria is grounded in a historical–cultural perspective that conceives psychological functions as dynamic processes whose genesis cannot be reduced either to the biological or to the individual level. Drawing on the anthropological studies of their time (
During the ontogenetic development of psychological functions, Vygotski noted that across different stages of development—and in their possible alterations—it is not the function itself that changes, but rather the relations that functions establish among themselves, giving rise to new groupings that did not exist at earlier stages. The formation of these new functional relations into novel groupings is what Vygotski termed Psychological Systems (PS) (
According to Luria (
The FS, developed by Anokhin in the 1930s, proposes a unit of neurophysiological integration capable of articulating PS with brain structure. Its integrative character lies in its ability to explain the general principles governing any system, among which the adaptive useful result stands out as the central organizing criterion. Unlike the reflex—whose endpoint is the action itself—and classical systemic conceptions based on reciprocal action among components, the FS is defined by the orientation of its processes toward adaptive functional results (
When the outcome of a functional action is insufficient, the system requires information about that outcome in order to reorganize the distribution of excitations in the appropriate direction. This informational process is referred to, within FS theory, as backward afferentation, which—unlike feedback in mechanical systems—is resolved internally through the system’s own regulatory mechanisms. This afferentation involves a returning sensory flow that is compared with an anticipatory regulatory configuration—the acceptor—and, in the case of discrepancy, initiates a new cycle of functional reorganization (
Moreover, because the organism operates within a continuous flow of result attainment, each achieved outcome immediately initiates the preparation of the next. In this process, the resolution of a given result configures the parameters of the subsequent one, generating an anticipatory functional template in memory, referred to as the acceptor. This acceptor fulfills a dual function: on the one hand, it acts as the receiver of the backward afferentation (feedback) generated by the action; on the other, it serves as an anticipatory model of the required result, against which incoming information is compared. When there is a match between the model and the feedback, the result is accepted and the system reorganizes functionally (
The self-organizing character of the FS is expressed through two complementary phases: afferent synthesis and efferent synthesis. Afferent synthesis precedes action and consists of the reciprocal integration of four decisive components: motivation, starting afferentation, contextual afferentation, and memory—understood as historically stabilized functional dispositions rather than stored representations. This interaction reduces the system’s indeterminacy and orients its dynamics toward a specific functional direction. It may involve reverberatory cycles among multiple neuronal elements and culminates in decision making, understood as a reduction of degrees of freedom and the selection of an action as a function of the dominant need, the organism’s experience, and the environmental context. Efferent synthesis, in turn, prepares motor execution through a further reduction of degrees of freedom, adjusting action patterns according to posture, spatial position, and proprioceptive information (
The characteristics described make it possible to understand that the useful adaptive result constitutes the organizing axis of the FS. Around this result, the functional architecture outlined above is articulated through a set of interrelated operations—conventionally described as eight moments—including afferent synthesis, decision making, generation of the acceptor of the action result, efferent synthesis, performance of the action, attainment of the result, backward afferentation, and comparison between the obtained result and its anticipated model. This architecture should not be understood as a rigid sequence of discrete states, but rather as an abstract functional organization that is dynamically implemented over time. From this perspective, the FS is not a reactive mechanism, but a dynamic, adaptive, and self-organizing system, capable of anticipating, reorganizing, and adjusting itself on the basis of its own results and the context in which it is embedded.
This architecture of the FS is articulated with the principles of PS in Luria’s conception of CFS. Starting from the premise that psychological activity and brain organization constitute a functional unity, Luria integrates the Vygotskian conception of PS—which emphasizes the social origin, systemic character, and dynamic localization of psychological functions—with Anokhin’s physiological theory of FS, which introduces the principles of adaptation, self-organization, result-oriented regulation, and anticipation. Within this synthesis, PS contribute the historical-cultural and mediated dimension of function, while FS provide the physiological framework that explains its realization and regulation within the central nervous system.
This theoretical convergence makes it possible to understand complex psychological processes not as activities localized in isolated structures, but as distributed functional systems that emerge from the dynamic interaction among cortical, subcortical, and social components. From this integration, Luria formulates three fundamental principles. First, CFS exhibit dynamic localization in the brain, undergoing functional reorganization as a function of the concrete conditions of the act, the task, and the context. Second, they are neither innate nor anatomically preformed, but develop throughout ontogeny through processes of social internalization, reflecting the historical and cultural conditions of the environment. Third, this development is mediated by signs and psychological instruments, with language playing a central role in the reorganization of the system’s functional relations (
The conceptions advanced by Vygotsky, Anokhin, and Luria converge on a shared characterization of psychological functions as dynamic functional organizations, historically constituted, goal-oriented, and sustained by variable relations among multiple components. Although this perspective is theoretically robust and clinically fruitful, it calls for a formal framework capable of accounting for functional invariance in the presence of variable realizations, adaptive reorganization across development and following brain injury, and the organizing role of signs and language in the system’s dynamics and topology. In this regard, topological dynamical systems theory offers particularly suitable conceptual and mathematical tools, as it allows the modeling of systems whose identity does not depend on fixed configurations, but rather on the preservation of invariant functional relations under dynamic transformations. The formalization of psychological and functional systems as Topological Functional Systems (TFS) thus constitutes a pathway for rigorously articulating the foundations of the historical–cultural approach with mathematical models capable of coherently supporting both theoretical explanation and the derivation of clinical implications in neuropsychology.
The characterization of psychological functions as Topological Functional Systems (TFS)—historically constituted, goal-oriented, and sustained by variable functional relations—raises the need for a formal framework capable of rigorously representing the coexistence of functional stability and variability of realizations. Indeed, if the identity of a function does not depend on fixed anatomical components or immutable operational sequences, but rather on the preservation of certain organizing principles that enable the attainment of a useful adaptive result under changing conditions, then static, localizationist, or purely structural models prove insufficient. The very logic of TFS theory calls for a formalization that can describe systems whose organization may transform without loss of identity, and that incorporate anticipation, feedback, and functional reorganization as a function of history and context. In this sense, formalization through tools from topological dynamical systems theory does not constitute an external extrapolation, but rather a natural consequence of the theoretical assumptions developed by Vygotsky, Anokhin, and Luria, insofar as it provides an appropriate language for modeling functional invariance, dynamic reorganization, and historical dependence—core features of psychological activity.
A TFS is proposed as a mathematically parsimonious and theoretically faithful formalization of the central thesis of FS theory: the identity of a function does not reside in a fixed anatomical configuration, but in the stability of the useful adaptive result that organizes the system (
The introduction of the space
From this perspective, signs and language—understood as primary sources of historical and social mediation of psychological activity—are not introduced as additional state variables, but rather as semiotic mechanisms of transversal organization of the system. This decision is consistent with the Vygotskian–Lurian tradition, according to which signs do not constitute specific functional contents, but instruments of regulation, anticipation, and reorganization of activity. Within the TFS model, signs modulate the functional core
From a mathematical and contemporary neuroscientific perspective, the “higher-order” character of
Accordingly, the dynamics of the TFS is formulated as a composition of continuous operators that implement the cybernetic cycle described by Anokhin (
In response to objections such as those raised by Pessoa (
A TFS is formalized as a topological dynamical system (
where the state space is defined as the product
with
The minimal functional space is defined as
where
which governs the processes of correction and functional reorganization of the system. Taken together, these variables encode the functional invariants of the TFS without fixing specific anatomical mechanisms, allowing its identity to be expressed as the preservation of the attainment of the useful result under dynamically variable realizations.
The factor
which encodes weights and/or intensities associated with interactions of different orders (edges, triads, 2-simplices, etc.) (
The dynamics of the TFS is specified primarily in a cycle-functional manner through a continuous update operator
whose iteration defines a discrete topological dynamical system
Here, S (afferent synthesis) integrates internal and contextual conditions, updating n and constraining η; D (decision making) selects a functional execution regime; A (acceptor) updates the anticipatory state a; P (efferent synthesis) configures execution parameters over η; U (action) deploys behavior; R (result) determines the attained functional state; B (backward afferentation) updates r with parameters of the obtained result; and C (comparison and correction) computes the discrepancy e = E(a, r) and feeds back onto n and η, including processes of plasticity and functional reorganization. In this formulation, the “components” of the functional system are not introduced as discrete phases of the state, but as the internal structure of the topological dynamics governing the joint evolution of the minimal functional state and the network support.
From a functional standpoint, the operators composing
These include S, D, A, and P. These transformations act endogenously on (
U and R correspond to the segment of effective interaction with the body and the environment. Topologically, this coupling is incorporated as part of the dynamics in
B and C close the cybernetic cycle, enabling discrepancy-guided correction and adaptive reorganization of the system. Dynamically, these operations are those that confer upon
The proposed formalization seeks to offer an initial mathematical framework for conceiving psychological functions as TFSs whose identity is defined by the stability of a useful adaptive result, rather than by fixed structural configurations. By distinguishing a minimal functional core, a variable network support, and an endogenous cybernetic dynamics, the TFS model makes it possible to describe, in a unified manner, anticipation, feedback, reorganization, and executional variability, which characterize both typical development and the transformations associated with brain injury or neurodivergence. This formalization is not intended to constitute a closed or exhaustive model, but rather a starting point open to refinement, extension, and empirical testing, in dialogue with future developments in both mathematics and neuroscience. Its main value lies in establishing a formal language coherent with the theoretical assumptions of the historical–cultural approach and with the theory of FS, thereby creating the conditions to explore, in the following section, its clinical implications for understanding symptoms, neuropsychological assessment, and functional reorganization.
In the effort to construct psychology as a general science grounded in the social, systemic, and dynamic character of psychological functions, Vygotsky proposed a methodological shift of paradigmatic scope. This shift is articulated in two central postulates: first, the replacement of analyses that fragment the system into parts—and thereby lose the emergent properties that only manifest through interaction—with an analysis based on units that preserve the essential properties of the whole; and second, the substitution of the dominant functional or structural analysis with an interfunctional or systemic analysis, oriented toward the study of the relations and connections among functions that are dynamically organized in the brain (
From this perspective, the TFS proposal can be understood as a contemporary and formalized extension of that program. At the research level, the foundational characteristics of TFS—the preservation of functional invariants under variable realizations, dynamic reorganization, and contextual dependence—respond directly to the Vygotskian requirement to identify units of analysis that preserve the logic of the system as a whole. In contrast to classical cognitive neuropsychology, where the method of double dissociation has been used to infer stable functional architectures and, in its strong reading, to support a modular ontology of cognition (
This methodological shift is accompanied by a profound transformation in the conception of the brain. In contrast to modular views, which assume relatively encapsulated and specialized components, contemporary neuroscience has converged on a view of the brain as an integrated dynamic system, in which functional segregation coexists with flexible and context-dependent processes of integration (
At the clinical level, this methodological and conceptual shift has direct consequences for the redefinition of both the symptom and neuropsychological rehabilitation. Within the Vygotskian–Lurian tradition, the symptom is not conceived as the direct loss of a function nor as the punctual reflection of a structural lesion, but rather as the manifestation of an alteration in the functional organization of the system, affecting the available modes of execution for achieving an adaptive result (
The clinical approach grounded in TFS, in this sense, is not oriented toward localizing deficits in isolated components, but toward analyzing patterns of functional reorganization, identifying which invariants are preserved, which modes of execution are compromised, and which new trajectories emerge as possible compensations. This perspective is not only consistent with the historical–cultural tradition, but also aligns with the core assumptions of TFS, offering a clinical framework capable of integrating assessment, diagnosis, and intervention within a genuinely systemic conception of the brain and psychological activity.
In a TFS-based clinical framework, the object of description and intervention is not a “function” conceived as an isolable component, but a functional system whose identity is defined by functional invariants (task, criterion of success, adaptive useful result) and whose realization admits multiple modes of execution through the reconfiguration of neural supports and strategies. This shift operationalizes the Vygotskian–Lurian thesis that the deficit is not primarily described as a “loss of function,” but as a reorganization of activity: orientation toward the task or result is preserved, while the dynamic conditions of its realization are altered, giving rise to syndromic profiles that depend on history and context (
A TFS-based clinical approach shifts the focus from “functions” conceived as isolable capacities toward units of functional organization defined by (i) invariants (task, criterion of success, and adaptive useful result) and (ii) variability of realization (multiple modes of execution and possible neural supports). This shift is consistent with the contemporary move away from discrete localizations toward circuits and causal networks as explanatory units of symptoms and syndromes: different lesions may converge on similar deficits when they disrupt the same functional circuit, which is precisely the type of “equivalence of realization” that TFS formalizes, rather than treating it as an anomaly of the model (
The first diagnostic level makes explicit the core identity of the system: which task/need organizes behavior, what the useful result is (criterion of success), and what the margin of tolerance is (cost/allowable variability before functional collapse). This specification supports a clinical diagnosis oriented toward participation, functioning, and goals, where the criterion of success is not “raising a score” but restoring meaningful functional performance. The tradition associated with Wilson (goal planning) and the evidence synthesized in reviews of cognitive rehabilitation support this view by arguing that the practical unit of intervention and evaluation should be the functional goal (and its attainment), rather than isolated test performance (
Within the TFS framework, the primary symptom is defined as the restriction or loss of a class of viable dynamic trajectories for achieving an adaptive useful result. This restriction does not imply the disappearance of the function as such, but rather the degradation of a specific mode of execution that previously allowed the function to be carried out in a stable and efficient manner. This mode of execution corresponds to a transfunctional operational principle that participates in the organization of multiple psychological activities. From the TFS perspective, this principle refers to the system’s capacity to sustain a given class of functionally equivalent trajectories associated with a specific type of control (for example, anticipation, sequencing, multimodal integration, or error regulation). The primary symptom emerges when this class of trajectories is lost or becomes unstable, reducing the repertoire of realizations available to the system.
This conception is consistent with contemporary network neuroscience, in which deficits are understood as consequences of perturbing functional circuits that support particular types of control, rather than as the “loss” of isolated cognitive modules. In this sense, approaches such as lesion network mapping provide strong empirical support for models in which different lesions can converge on similar symptoms by affecting common circuits, and in which the same function can be maintained through alternative realizations under conditions of functional reorganization (
The third level characterizes the systemic effects of reorganization: costs (time/effort), new contextual dependencies, reduced generalization, fatigue, and compensatory strategies that may appear as “additional deficits” when interpreted from a modular perspective. This level connects both with Luria’s syndromic clinical approach and with contemporary observations of recovery understood as system reconfiguration rather than point-by-point restitution. One advantage of the TFS framework is that it integrates these phenomena as constitutive elements of diagnosis, instead of treating them as noise or nonspecific comorbidity.
At this level, TFS diagnosis characterizes the systemic effects derived from the functional reorganization that the system implements after the loss or degradation of a dominant mode of execution. These effects do not constitute independent deficits, but rather structural costs of the new dynamic regime through which the system attempts to preserve the adaptive useful result. Clinically, they manifest as increased time and effort, greater dependence on contextual cues or external supports, reduced generalization, functional fatigue, and the emergence of specific compensatory strategies. When read through a modular lens, these phenomena may be mistakenly interpreted as “additional deficits” or comorbidities; however, from the TFS perspective, they are understood as necessary consequences of the global reorganization of the system following the degradation of a mode of execution.
This interpretation is fully consistent with Luria’s syndromic clinical approach, in which secondary and tertiary symptoms reflect the redistribution of functional activity, as well as with contemporary rehabilitation frameworks that conceive recovery not as point-by-point restitution of damaged components, but as adaptive reconfiguration of distributed systems, with functional gains accompanied by inevitable trade-offs (
The minimal core X = (n, a, r) makes it possible to translate clinical reasoning into observable variables without resorting to a modular catalog of “components.” The dominant need n(t) functions as a clinical prioritization vector (which goals or needs govern the organization of performance), anticipation a(t) reflects the quality of prospective control (preparation, functional set, planning), and the return signal associated with the actually achieved result r(t) captures the quality of monitoring and error-based updating. The discrepancy e(t) = E(a(t), r(t)) operationalizes an interpretable “clinical error.” This articulation allows heterogeneous clinical presentations to be described with a parsimonious language: not damaged “modules,” but dominant patterns of dysregulation (anticipation, monitoring, prioritization) that are also directly translatable into therapeutic targets (anticipatory regulation, improving feedback use, reorganizing goals). This bridge avoids the inferential leap from “dissociation → module,” showing instead how dissociations can emerge from reorganizations of control within distributed systems.
The TFS model requires that clinical assessment be structured as a topological–functional profile rather than as an aggregation of isolated scores. This profile integrates, in a coordinated manner: (i) the degree of attainment of the adaptive useful result, (ii) the costs associated with its achievement (time, effort, contextual dependence, fatigue), (iii) the stability, variability, and flexibility of the available execution trajectories, and (iv) the resulting syndromic configuration, understood as the articulation between primary symptoms, systemic effects, and compensatory strategies. This approach is consistent with contemporary goal- and person-centered rehabilitation practices, including methodologies such as Goal Attainment Scaling, which allow the operationalization of individualized functional goal achievement without sacrificing methodological rigor or comparability.
From the TFS perspective, this profile cannot be adequately assessed without recognizing the mediating role of signs and language in the regulation of functional activity. Instructions, self-verbalization, symbolic supports, and external cues are not treated as “methodological aids” that distort measurement, but as structural indicators of the system’s functional regime. Dependence on semiotic mediation, its effectiveness in stabilizing trajectories, and its impact on performance generalization provide key diagnostic information about the organization, costs, and trade-offs of the functional system. In this sense, the contemporary shift toward ecological and digital assessments—together with critiques of clinical inferences based exclusively on decontextualized tests—opens a methodological framework particularly well suited to operationalizing trajectories, costs, and semiotic dependence, precisely the constructs that the TFS model places at the center of clinical analysis (
Within the TFS framework, rehabilitation is defined by two functionally coupled objectives: (i) preserving or restoring the adaptive useful result and (ii) expanding the repertoire of viable trajectories that allow this result to be achieved with a more favorable cost–benefit balance. This formulation is fully consistent with contemporary goal-oriented neuropsychological rehabilitation focused on everyday functioning and on collaboration among patient, family, and therapeutic team, as consistently defended in the clinical literature (
Operationally, interventions can be organized across three interdependent levels: (I) interventions targeting the functional core X, aimed at recalibrating result anticipation, improving monitoring and error use, and modulating motivational priorities; (II) interventions targeting N, directed at inducing new execution routes, stabilizing functional configurations, and promoting adaptive generalization; and (III) interventions targeting task–context coupling, through the ecological design of activities, difficulty gradients, and contextual manipulation. Across all these levels, signs and language play a central role as instruments of regulation and functional reorganization. Semiotic mediation—through instructions, narratives, self-instructions, explicit rules, and symbolic supports—acts as a clinical operator that enables the modification of priorities, the stabilization of trajectories, the selection of functional configurations, and the reduction of costs without requiring the restoration of damaged components.
From this perspective, language is not conceived as an additional functional domain, but as a transversal instrument of control, anticipation, and compensation, consistent with the Vygotskian–Lurian tradition and with the cybernetic logic of the functional system. TFS rehabilitation therefore does not seek to eliminate semiotic dependence, but to optimize its use, evaluating when it stabilizes functioning, when it introduces undesirable trade-offs, and when it can be stabilized as an endogenous regulatory routine to foster a more autonomous and flexible reorganization of the system.
Rather than treating double dissociation as near-direct evidence for modules, the TFS approach reinterprets it as a possible consequence of: (i) a change in the control regime within X (anticipation/monitoring/prioritization), (ii) alterations in recruitment within N, and (iii) compensatory reorganization accompanied by trade-offs. This position is not merely “philosophical.” There is a substantial classical literature showing that double dissociations can emerge in distributed systems (e.g., connectionist networks) without implying separable components in a strong modular sense, as well as methodological critiques of the inference “dissociation → module” (
The mathematical formalization is not introduced as an ornament, but as a tool to: (i) define invariants and realization equivalences with precision, (ii) model and compare functional regimes (e.g., different X profiles with different costs), and (iii) clinically link symptoms to constraints on trajectories and to changes in recruitment within N. In dialogue with contemporary circuit-based neuroscience, this formalization is compatible with the agenda of moving from “anatomical” descriptions to causal descriptions at the circuit level (
In practical terms, the formalism opens three avenues: (1) simulation of intervention scenarios (how changes in anticipation/monitoring or in the stability of network configurations alter functional cost), (2) qualitative prediction of trade-offs (which systemic symptoms are expected given a primary constraint), and (3) the design of dynamics-oriented assessment (which tasks/conditions reveal regime shifts and which measures capture invariants versus costs). This approach also aligns with contemporary trends in neuropsychology toward more integrated and technologically instrumentable frameworks (digital assessment, longitudinal metrics), without collapsing into a reductionism in which “the test is everything” (
The present work set out to substantiate a topological formalization of FS that is, at the same time, theoretically congruent with the historical–cultural and neuropsychological tradition inaugurated by Vygotsky and Luria, mathematically consistent with topological dynamics and cybernetics, and clinically pertinent for the understanding and rehabilitation of psychological functions. In this sense, TFS do not constitute a merely descriptive alternative model, but rather an integrative proposal aimed at clarifying the organization, variability, and functional stability of psychological activity under normal conditions, atypical development, brain injury, or psychopathology.
Contemporary neuropsychology has developed multiple frameworks to explain the relationship between brain organization, psychological activity, and clinical symptomatology. In schematic terms, classical and cognitive approaches have shared two central assumptions: (i) a tendency to describe psychological functions as relatively discrete entities—either anatomically localized (
From this perspective, the TFS model aligns with contemporary developments that conceive the brain as a dynamic, distributed, multilevel system, organized simultaneously by principles of integration and segregation (
One of the central contributions of the TFS approach is the clarification of the status of functional variability. Rather than interpreting variability as noise, secondary compensation, or failure of control, variability is conceived as a structural property of the system, compatible with the preservation of functional invariants. This idea, already present in Bernstein’s theory of movement and in Anokhin’s theory of functional systems, acquires here an explicit formalization: functional identity is not defined by fixed configurations, but by the preservation of the result across multiple dynamic trajectories.
The introduction of a minimal functional state, organized around need, anticipation, and feedback, further allows for a natural integration of the anticipatory character of psychological activity. At this point, the model enters into dialogue with contemporary predictive approaches—including Bayesian brain models and allostatic formulations (
At the clinical level, the TFS model proposes a systematic reinterpretation of the neuropsychological symptom. Instead of conceiving it as the loss of a function or an isolable component, the primary symptom is defined as the degradation of a mode of execution, that is, as a restriction of the set of functionally equivalent trajectories available to achieve a useful adaptive result. This degradation does not imply the disappearance of the goal of activity, but rather the loss of stability, efficiency, or viability of a specific class of realizations that previously organized performance. From this initial alteration, systemic symptoms emerge as cascading effects of the global reorganization of the system, including compensatory strategies, functional trade-offs, and new contextual dependencies, which must be interpreted as structural consequences of the new functional regime rather than as additional independent deficits.
This reading allows for an explicit articulation between the syndromic logic developed by Luria and contemporary evidence showing that different lesions can converge on similar functional profiles, and that recovery depends less on anatomical restitution than on the reorganization of networks and strategies. It also provides a coherent framework for goal-oriented rehabilitation, consistent with current models focused on participation, ecological functioning, and quality of life, but with a more explicit theoretical and mathematical grounding regarding what is modified, what is preserved, and how change should be evaluated.
At this point, it is necessary to clarify the status of topology in the TFS proposal. Unlike other contemporary approaches that employ topological tools—such as higher-order networks, simplicial complexes, or derived metrics (
In this sense, developments such as higher-order networks or simplicial complexes are naturally integrated into the space of functional configurations of the system, but are subsumed within a broader framework in which cybernetic dynamics, anticipation, and semiotic mediation organize activity. The TFS proposal therefore does not merely apply topology to neuroscience, but advances a neuropsychology whose functional ontology is already topological.
This emphasis makes it possible to extend the scope of the model beyond strictly neuropsychological clinical contexts to psychological practice, psychiatry, and education, in coherence with the Vygotskian conception of psychological functions as socially originated and semiotically mediated processes. Within this framework, the mathematical formalization of TFS is not an end in itself, but a tool to clarify concepts, simulate functional dynamics, explore scenarios of reorganization, and ultimately guide clinical and pedagogical decisions in a more systematic manner.
TFS are proposed as an open research program, susceptible to refinement, empirical testing, and expansion, but with a theoretical, mathematical, and clinical foundation sufficiently robust to contribute meaningfully to the contemporary development of a genuinely systemic neuropsychology. Ultimately, we remain convinced that “systems” and “finality” must continue to be the alpha and the omega of our most immediate scientific endeavor.
JG: Conceptualization, Data curation, Formal Analysis, Funding acquisition, Investigation, Methodology, Project administration, Resources, Software, Supervision, Validation, Visualization, Writing – original draft, Writing – review & editing.
The author(s) declared that this work was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.
The author(s) declared that generative AI was used in the creation of this manuscript. The author acknowledges the assistance of ChatGPT (GPT-5, OpenAI, 2025 version), which was used for translation and for the verification of certain mathematical formulas. The final content was manually reviewed, validated, and edited by the author prior to inclusion in the document.
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