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Complex processes from dynamical architectures with time-scale hierarchy.

Perdikis D, Huys R, Jirsa V - PLoS ONE (2011)

Bottom Line: Thereto, we build on the (phase) flow of a system, which prescribes the temporal evolution of its state variables.The phase flow topology allows for the unambiguous classification of qualitatively distinct processes, which we consider to represent the functional units or modes within the dynamical architecture.We reveal a tradeoff of the interactions between internal and external influences, which offers a theoretical justification for the efficient composition of complex processes out of non-trivial elementary processes or functional modes.

View Article: PubMed Central - PubMed

Affiliation: Theoretical Neuroscience Group, UMR6233 Institut Science du Mouvement, University of the Mediterranean, Marseille, France. dionysios.perdikis@etumel.univmed.fr

ABSTRACT
The idea that complex motor, perceptual, and cognitive behaviors are composed of smaller units, which are somehow brought into a meaningful relation, permeates the biological and life sciences. However, no principled framework defining the constituent elementary processes has been developed to this date. Consequently, functional configurations (or architectures) relating elementary processes and external influences are mostly piecemeal formulations suitable to particular instances only. Here, we develop a general dynamical framework for distinct functional architectures characterized by the time-scale separation of their constituents and evaluate their efficiency. Thereto, we build on the (phase) flow of a system, which prescribes the temporal evolution of its state variables. The phase flow topology allows for the unambiguous classification of qualitatively distinct processes, which we consider to represent the functional units or modes within the dynamical architecture. Using the example of a composite movement we illustrate how different architectures can be characterized by their degree of time scale separation between the internal elements of the architecture (i.e. the functional modes) and external interventions. We reveal a tradeoff of the interactions between internal and external influences, which offers a theoretical justification for the efficient composition of complex processes out of non-trivial elementary processes or functional modes.

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Overview of the four functional architectures.Each column represents a functional architecture with time scale separation as indicated at the top. Lower row: Time series depicting the operational signals σi,j(t) (lower graph; i, j index the system's dimensions –in columns 1 and 2- or fingers –in column 3- where σ operates upon) and the system's output x1,3(t) (upper graph; state variables accounting for position). Blue and green lines represent σi and x1 versus σj and x3, respectively. Upper rows: The time evolution is indicated by the arrows. Each square panel in the upper rows represents the phase space of a particular functional mode; sequential panels (in time) indicate changes in the functional modes; dotted lines indicate the persistence of a particular mode (until substituted by another one). Paired panels (left, right) represent the modes corresponding to finger 1 and 2, respectively (except for the fourth column where the two fingers are coupled and where only three out of the four dimensions of the system's phase space can be shown). From left to right; first column (τσ≪τf): σ2,4 provide instantaneous functional kicks to the modes (see equation (2)); second column (τσ≈τf): fixed points are driven by σ1,3 through phase space (one movement cycle depicted only –also see equation (3)); third column (τσ≫τf): σ1,2 sequentially select distinct functional modes (see equation (5)); fourth column (τσ→∞): σ = constant (has no effect), i.e., the system is entirely autonomous. Notice that the more the time scales of the operational signals and the functional modes differ, the more the role of the operational signals decreases and the complexity of the phase flows involved increases.
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pone-0016589-g003: Overview of the four functional architectures.Each column represents a functional architecture with time scale separation as indicated at the top. Lower row: Time series depicting the operational signals σi,j(t) (lower graph; i, j index the system's dimensions –in columns 1 and 2- or fingers –in column 3- where σ operates upon) and the system's output x1,3(t) (upper graph; state variables accounting for position). Blue and green lines represent σi and x1 versus σj and x3, respectively. Upper rows: The time evolution is indicated by the arrows. Each square panel in the upper rows represents the phase space of a particular functional mode; sequential panels (in time) indicate changes in the functional modes; dotted lines indicate the persistence of a particular mode (until substituted by another one). Paired panels (left, right) represent the modes corresponding to finger 1 and 2, respectively (except for the fourth column where the two fingers are coupled and where only three out of the four dimensions of the system's phase space can be shown). From left to right; first column (τσ≪τf): σ2,4 provide instantaneous functional kicks to the modes (see equation (2)); second column (τσ≈τf): fixed points are driven by σ1,3 through phase space (one movement cycle depicted only –also see equation (3)); third column (τσ≫τf): σ1,2 sequentially select distinct functional modes (see equation (5)); fourth column (τσ→∞): σ = constant (has no effect), i.e., the system is entirely autonomous. Notice that the more the time scales of the operational signals and the functional modes differ, the more the role of the operational signals decreases and the complexity of the phase flows involved increases.

Mentions: The operational signal σ(t) operates (upon) the functional modes and generally will not be independent of xi. Here we wish to focus on the causal effects of the operational signal upon the functional modes. Let τf and τσ denote the time scales corresponding to a particular functional mode and operational signal σ(t) respectively. For different functional architectures, τσ may operate on various time scales relative to τf and could in principle span a continuum of scales. Here, we choose four different instantiations of time scale separations (see Figure 3 for an overview). In cases in which σ(t) acts much faster than the functional mode (i.e., τσ≪τf), σ(t) operates upon the mode (exemplified below as Scenario 1). In those cases where σ(t) acts on a time scale similar to that of the functional mode (i.e., τσ≈τf−Scenario 2), σ(t) may be said to operate the functional mode. In Scenario 3 we consider the case where σ(t) acts much slower than the functional mode (τσ≪τf). Finally, in the fourth architecture σ(t) can be considered as time-independent (i.e., σ(t)≈ constant during the functional process or equivalently τσ→∞−Scenario 4). All scenarios are exemplified below.


Complex processes from dynamical architectures with time-scale hierarchy.

Perdikis D, Huys R, Jirsa V - PLoS ONE (2011)

Overview of the four functional architectures.Each column represents a functional architecture with time scale separation as indicated at the top. Lower row: Time series depicting the operational signals σi,j(t) (lower graph; i, j index the system's dimensions –in columns 1 and 2- or fingers –in column 3- where σ operates upon) and the system's output x1,3(t) (upper graph; state variables accounting for position). Blue and green lines represent σi and x1 versus σj and x3, respectively. Upper rows: The time evolution is indicated by the arrows. Each square panel in the upper rows represents the phase space of a particular functional mode; sequential panels (in time) indicate changes in the functional modes; dotted lines indicate the persistence of a particular mode (until substituted by another one). Paired panels (left, right) represent the modes corresponding to finger 1 and 2, respectively (except for the fourth column where the two fingers are coupled and where only three out of the four dimensions of the system's phase space can be shown). From left to right; first column (τσ≪τf): σ2,4 provide instantaneous functional kicks to the modes (see equation (2)); second column (τσ≈τf): fixed points are driven by σ1,3 through phase space (one movement cycle depicted only –also see equation (3)); third column (τσ≫τf): σ1,2 sequentially select distinct functional modes (see equation (5)); fourth column (τσ→∞): σ = constant (has no effect), i.e., the system is entirely autonomous. Notice that the more the time scales of the operational signals and the functional modes differ, the more the role of the operational signals decreases and the complexity of the phase flows involved increases.
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Related In: Results  -  Collection

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getmorefigures.php?uid=PMC3037373&req=5

pone-0016589-g003: Overview of the four functional architectures.Each column represents a functional architecture with time scale separation as indicated at the top. Lower row: Time series depicting the operational signals σi,j(t) (lower graph; i, j index the system's dimensions –in columns 1 and 2- or fingers –in column 3- where σ operates upon) and the system's output x1,3(t) (upper graph; state variables accounting for position). Blue and green lines represent σi and x1 versus σj and x3, respectively. Upper rows: The time evolution is indicated by the arrows. Each square panel in the upper rows represents the phase space of a particular functional mode; sequential panels (in time) indicate changes in the functional modes; dotted lines indicate the persistence of a particular mode (until substituted by another one). Paired panels (left, right) represent the modes corresponding to finger 1 and 2, respectively (except for the fourth column where the two fingers are coupled and where only three out of the four dimensions of the system's phase space can be shown). From left to right; first column (τσ≪τf): σ2,4 provide instantaneous functional kicks to the modes (see equation (2)); second column (τσ≈τf): fixed points are driven by σ1,3 through phase space (one movement cycle depicted only –also see equation (3)); third column (τσ≫τf): σ1,2 sequentially select distinct functional modes (see equation (5)); fourth column (τσ→∞): σ = constant (has no effect), i.e., the system is entirely autonomous. Notice that the more the time scales of the operational signals and the functional modes differ, the more the role of the operational signals decreases and the complexity of the phase flows involved increases.
Mentions: The operational signal σ(t) operates (upon) the functional modes and generally will not be independent of xi. Here we wish to focus on the causal effects of the operational signal upon the functional modes. Let τf and τσ denote the time scales corresponding to a particular functional mode and operational signal σ(t) respectively. For different functional architectures, τσ may operate on various time scales relative to τf and could in principle span a continuum of scales. Here, we choose four different instantiations of time scale separations (see Figure 3 for an overview). In cases in which σ(t) acts much faster than the functional mode (i.e., τσ≪τf), σ(t) operates upon the mode (exemplified below as Scenario 1). In those cases where σ(t) acts on a time scale similar to that of the functional mode (i.e., τσ≈τf−Scenario 2), σ(t) may be said to operate the functional mode. In Scenario 3 we consider the case where σ(t) acts much slower than the functional mode (τσ≪τf). Finally, in the fourth architecture σ(t) can be considered as time-independent (i.e., σ(t)≈ constant during the functional process or equivalently τσ→∞−Scenario 4). All scenarios are exemplified below.

Bottom Line: Thereto, we build on the (phase) flow of a system, which prescribes the temporal evolution of its state variables.The phase flow topology allows for the unambiguous classification of qualitatively distinct processes, which we consider to represent the functional units or modes within the dynamical architecture.We reveal a tradeoff of the interactions between internal and external influences, which offers a theoretical justification for the efficient composition of complex processes out of non-trivial elementary processes or functional modes.

View Article: PubMed Central - PubMed

Affiliation: Theoretical Neuroscience Group, UMR6233 Institut Science du Mouvement, University of the Mediterranean, Marseille, France. dionysios.perdikis@etumel.univmed.fr

ABSTRACT
The idea that complex motor, perceptual, and cognitive behaviors are composed of smaller units, which are somehow brought into a meaningful relation, permeates the biological and life sciences. However, no principled framework defining the constituent elementary processes has been developed to this date. Consequently, functional configurations (or architectures) relating elementary processes and external influences are mostly piecemeal formulations suitable to particular instances only. Here, we develop a general dynamical framework for distinct functional architectures characterized by the time-scale separation of their constituents and evaluate their efficiency. Thereto, we build on the (phase) flow of a system, which prescribes the temporal evolution of its state variables. The phase flow topology allows for the unambiguous classification of qualitatively distinct processes, which we consider to represent the functional units or modes within the dynamical architecture. Using the example of a composite movement we illustrate how different architectures can be characterized by their degree of time scale separation between the internal elements of the architecture (i.e. the functional modes) and external interventions. We reveal a tradeoff of the interactions between internal and external influences, which offers a theoretical justification for the efficient composition of complex processes out of non-trivial elementary processes or functional modes.

Show MeSH
Related in: MedlinePlus