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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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Related in: MedlinePlus

Illustration of Scenario 2.Scenario 2 (see equation (3)) shows a sketch of the phase flows (linear point attractor -panel A) as well as the output time series (positions x1,3 and operational signals σ1,3(t) -panel B). Colour coding and fixed point notation are the same as in the previous figure. A single pulse of σ1(t) and its effect on the phase flow of the first finger are blown up in panel A, depicting five characteristic instances of the phase flow. The phase flows change at the same time scale as the functional process (τσ≈τf), since the position of the attracting equilibrium point is constantly assigned by the operational signal.
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pone-0016589-g005: Illustration of Scenario 2.Scenario 2 (see equation (3)) shows a sketch of the phase flows (linear point attractor -panel A) as well as the output time series (positions x1,3 and operational signals σ1,3(t) -panel B). Colour coding and fixed point notation are the same as in the previous figure. A single pulse of σ1(t) and its effect on the phase flow of the first finger are blown up in panel A, depicting five characteristic instances of the phase flow. The phase flows change at the same time scale as the functional process (τσ≈τf), since the position of the attracting equilibrium point is constantly assigned by the operational signal.

Mentions: The functional modes implemented below consist of 4-dimensional phase flows (two dimensions per effector). In all cases, (x1, x2, x3, x4), are the state variables of the system, T1,2 are the effectors' main time constants while k1,2 introduce a time scale separation between the state variables of each effector's phase flow. Thus, (x1, x2), T1, k1 and (x3, x4), T2, k2 refer are associated with the first and second effector respectively. At the same time, the state variables (x1, x3) correspond to the effectors' positions, while (x2, x4) correspond to their velocities. This notation is used in Figures 4, 5, 6 and 7 describing the results and in the presentation of Scenario 4 below. The same notation is used for the operational signals where the indexes of σ correspond to either the equation's state variables (1 to 4 in Scenarios 1 and 2) or to an effector (1 or 2, in Scenario 3). For reasons of brevity, we only present the two-dimensional phase flows used to model either both or each one of the effectors for Scenarios 1–3 below, since the two effectors are modeled as uncoupled (and can thus be presented separately). The 4-dimensional system in Scenario 4 is presented entirely as its corresponding two effectors are coupled. No claim for the generating mechanisms of the operational signals is made in the present work. The ones used in the simulations where chosen such as that the resulting multidimensional operational signals are non-autonomous and their different dimensions are uncorrelated. All simulations were carried out in MATLAB, while a Runge-Kutta algorithm of 4th order has been used for the integration of the dynamical systems. Further details on the models and simulations can be found in the Supporting Information (Text S1).


Complex processes from dynamical architectures with time-scale hierarchy.

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

Illustration of Scenario 2.Scenario 2 (see equation (3)) shows a sketch of the phase flows (linear point attractor -panel A) as well as the output time series (positions x1,3 and operational signals σ1,3(t) -panel B). Colour coding and fixed point notation are the same as in the previous figure. A single pulse of σ1(t) and its effect on the phase flow of the first finger are blown up in panel A, depicting five characteristic instances of the phase flow. The phase flows change at the same time scale as the functional process (τσ≈τf), since the position of the attracting equilibrium point is constantly assigned by the operational signal.
© Copyright Policy
Related In: Results  -  Collection

Show All Figures
getmorefigures.php?uid=PMC3037373&req=5

pone-0016589-g005: Illustration of Scenario 2.Scenario 2 (see equation (3)) shows a sketch of the phase flows (linear point attractor -panel A) as well as the output time series (positions x1,3 and operational signals σ1,3(t) -panel B). Colour coding and fixed point notation are the same as in the previous figure. A single pulse of σ1(t) and its effect on the phase flow of the first finger are blown up in panel A, depicting five characteristic instances of the phase flow. The phase flows change at the same time scale as the functional process (τσ≈τf), since the position of the attracting equilibrium point is constantly assigned by the operational signal.
Mentions: The functional modes implemented below consist of 4-dimensional phase flows (two dimensions per effector). In all cases, (x1, x2, x3, x4), are the state variables of the system, T1,2 are the effectors' main time constants while k1,2 introduce a time scale separation between the state variables of each effector's phase flow. Thus, (x1, x2), T1, k1 and (x3, x4), T2, k2 refer are associated with the first and second effector respectively. At the same time, the state variables (x1, x3) correspond to the effectors' positions, while (x2, x4) correspond to their velocities. This notation is used in Figures 4, 5, 6 and 7 describing the results and in the presentation of Scenario 4 below. The same notation is used for the operational signals where the indexes of σ correspond to either the equation's state variables (1 to 4 in Scenarios 1 and 2) or to an effector (1 or 2, in Scenario 3). For reasons of brevity, we only present the two-dimensional phase flows used to model either both or each one of the effectors for Scenarios 1–3 below, since the two effectors are modeled as uncoupled (and can thus be presented separately). The 4-dimensional system in Scenario 4 is presented entirely as its corresponding two effectors are coupled. No claim for the generating mechanisms of the operational signals is made in the present work. The ones used in the simulations where chosen such as that the resulting multidimensional operational signals are non-autonomous and their different dimensions are uncorrelated. All simulations were carried out in MATLAB, while a Runge-Kutta algorithm of 4th order has been used for the integration of the dynamical systems. Further details on the models and simulations can be found in the Supporting Information (Text S1).

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