Mathematical models of central pattern generators in locomotion: III. Interlimb model for the cat

Possible neural connective patterns and functions with respect to interlimb coordination are studied theoretically with a mathematical model of the central pattern generating system for cat locomotion. Activities in populations of neurons controlling limb joint flexors and extensors in all four limbs are represented by a system of nonlinear differential equations. Solutions of the system for various parameter values simulate various gaits of the cat. The model is shown to be capable of generating all gaits of the cat and accounting for corresponding phase changes in interlimb coordination. The model also exhibits smooth changes of gait, and smooth initiation and termination of stepping. Further, within each limb, muscle sequencing, step cycle phases, and flexor-extensor interactions can be studied. The model suggests that one of the simplest mechanisms for a central command system to change the gait is via inhibition of specific interlimb propriospinal pathways. In a final section, properties of both proposed single limb and interlimb models are reviewed with specific reference to planning future experimental and theoretical studies.     

Mathematical models of central pattern generators in locomotion: I. Current problems

As a background for subsequent studies of mathematical models of central pattern generators in locomotion (Stafford & Barnwell, 1985a, b) relevant aspects of the literature on locomotion are reviewed, concepts of locomotion discussed, and extant models considered. Advantages and disadvantages of present models are discussed, and the need for mathematical models is emphasized. It is shown that realistic models of pattern generation in locomotion must take numerous factors into account, including phases of step cycle, muscle sequencing, gait and interlimb coordination, initiation and cessation of locomotion, and many aspects of neuromuscular control and function.     

Mathematical models of foreign policy decision-making: Compensatory vs. noncompensatory

There are presently two leading foreign policy decision-making paradigms in vogue. The first is based on the classical or rational model originally posited by von Neumann and Morgenstern to explain microeconomic decisions. The second is based on the cybernetic perspective whose groundwork was laid by Herbert Simon in his early research on bounded rationality. In this paper we introduce a third perspective — thepoliheuristic theory of decision-making — as an alternative to the rational actor and cybernetic paradigms in international relations. This theory is drawn in large part from research on heuristics done in experimental cognitive psychology. According to the poliheuristic theory, policy makers use poly (many) heuristics while focusing on a very narrow range of options and dimensions when making decisions. Among them, the political dimension is noncompensatory. The paper also delineates the mathematical formulations of the three decision-making models.The authors thank Raymond Dacey for his helpful comments.     

II. Mathematical modelling of election predictions: Comments to Simon's reply

Herbert A. Simon's reply (Inquiry, Vol. 25, No. 3) to my criticism of his 1954 paper is not to the point. He fails to respond to some of my arguments and misconceives others. One of his misconceptions is that any mathematical deduction from empirical premises which are formulated mathematically will necessarily lead to empirically valid conclusions. This claim is particularly unwarrantable in Simon's case since his mathematical premise, the continuity of the reaction function, is empirically meaningless.     

Mathematical formulation of multivariate euclidean models for discrimination methods

Multivariate models for the triangular and duo-trio methods are described in this paper. In both cases, the mathematical formulation of Euclidean models for these methods is derived and evaluated for the bivariate case using numerical quadrature. Theoretical results are compared with those obtained using Monte Carlo simulation which is validated by comparison with previously published theoretical results for univariate models of these methods. This work is discussed in light of its importance to the development of a new theory for multidimensional scaling in which the traditional assumption can be eliminated that proximity measures and perceptual distances are monotonically related.     

Mathematical models and the experimental analysis of behavior

The use of mathematical models in the experimental analysis of behavior has increased over the years, and they offer several advantages. Mathematical models require theorists to be precise and unambiguous, often allowing comparisons of competing theories that sound similar when stated in words. Sometimes different mathematical models may make equally accurate predictions for a large body of data. In such cases, it is important to find and investigate situations for which the competing models make different predictions because, unless two models are actually mathematically equivalent, they are based on different assumptions about the psychological processes that underlie an observed behavior. Mathematical models developed in basic behavioral research have been used to predict and control behavior in applied settings, and they have guided research in other areas of psychology. A good mathematical model can provide a common framework for understanding what might otherwise appear to be diverse and unrelated behavioral phenomena. Because psychologists vary in their quantitative skills and in their tolerance for mathematical equations, it is important for those who develop mathematical models of behavior to find ways (such as verbal analogies, pictorial representations, or concrete examples) to communicate the key premises of their models to nonspecialists.     

Predictability of locomotion: Effects on updating of spatial situation models during narrative comprehension

We investigated how the updating of spatial situation models during narrative comprehension depends on the interaction of cognitive abilities and text characteristics. Participants with low verbal and visuospatial abilities and participants with high abilities read narratives in which the protagonist's motions in a fictitious building were either highly predictable or very hard to predict. In Experiment 1, high-ability readers updated spatial situation models only if the protagonist's motions were hard to predict, whereas low-ability readers did so only if the motions were highly predictable. In Experiment 2, facilitated integrative spatial processing compensated for the low-ability participants' resource limitations. As a result, both ability groups updated spatial situation models with hard-to-predict protagonist motions. These results highlight the interactions of cognitive abilities and text characteristics in spatial situation model updating.     

Mathematical models for coordinated interpersonal timing in mother-infant interactions in the first year of life

Two mathematical models of social contingency are examined in terms of their development over the first year of life. The interactions of 53 mothers and their infants were recorded at 6 weeks, 4 months, and 12 months. The infants' gazes at 6 weeks, the mothers' vocal behavior at 6 weeks, and the vocal behavior of the mother and infant at 4 and 12 months were automatically coded in terms of four states. The conditional dependence model and the response effects model were computed for each interaction at each age, and the coefficients of the models were examined as a function of age. The relative success of the models as estimates of moment-to-moment contingency as well as their variations with age are discussed.     

Use of central and peripheral optical flow in stance and locomotion in young walkers

Young walkers (up to 5 years of age) were presented with optical flow in a moving room. Flow was global or was restricted to either the center or the periphery of the visible optic array. On standing trials the response rate was greatest when peripheral flow was available. The availability of central flow had a smaller effect on standing, and the younger children showed greater response rates to frontal flow than did the older ones. There was a strong negative correlation between age and response rate for all conditions. Flow also affected stability during locomotion. Response rate was again related to the location of the available flow. It is concluded that children show the same relative sensitivity for flow in the periphery of the dynamic structure of the optic array as has been observed in adults, but that this differentiation of different areas of optical structure is not yet fully developed when children learn to stand.     

Acquisition of locational information about reference points during locomotion: The role of central information processing

The role of central information processing for the acquisition of information about locations in a large-scale environment was investigated in two experiments. Subjects with and without a concurrent backwards counting task traversed a locomotion path repeatedly attempting to learn the locations of six reference points designated along the path, then estimated direction and distance numerically to these reference points when traversing the same path further times in a subsequent test phase. The main dependent measures were the latencies and the constant and variable errors of the estimates. The results indicated that acquisition of information about the locations was disrupted when central information processing was interfered with. However, central processing seemed to be less critical for long-term storage of information about the locomotion path. The latter type of information could thus be used by the subjects with the concurrent task when estimating the locations. Finally, the results suggested that information about the path was used also by the subjects without any concurrent task when the test phase required them to remain oriented relative to several reference points at the same time.     

Visual control of locomotion: strategies for changing direction and for going over obstacles.

Dynamics of gait adjustments required to go over obstacles and to alter direction of locomotion when cued visually were assessed through the measurement of ground reaction forces, muscle activity, and kinematics. The time of appearance of obstacles of varying heights, their position within the step cycle, and cue lights for direction change were varied. Direction change must be planned in the previous step to reduce the acceleration of the body center of mass toward the landing foot to 0. The inability of steering within the step cycle is due to the incapacity of muscles to rotate the body and translate it along the mediolateral axes. For obstacle avoidance, Ss systematically manipulated the gait patterns as a function of obstacle height and position and the time available within the ongoing step. Greater supraspinal involvement in control of locomotion is found.     

From central pattern generator to sensory template in the evolution of birdsong

Central nervous networks, be they a part of the human brain or a group of neurons in a snail, may be designed to produce distinct patterns of movement. Central pattern generators can account for the development and production of normal vocal signals without auditory feedback in non-songbirds. Songbirds need auditory feedback to develop and maintain the normal song of their species. The prerequisite for the use of auditory feedback for the control of song is a set of acoustic criteria or a template to which voice must match. The template method perhaps evolved to free birds from fixed central pattern generators, resulting in the evolution of diverse and complex songs among oscine songbirds. The evolution of human speech may have followed a similar course.     

Visual pattern memory requires foraging function in the central complex of Drosophila

The role of the foraging (for) gene, which encodes a cyclic guanosine-3',5'-monophosphate (cGMP)-dependent protein kinase (PKG), in food-search behavior in Drosophila has been intensively studied. However, its functions in other complex behaviors have not been well-characterized. Here, we show experimentally in Drosophila that the for gene is required in the operant visual learning paradigm. Visual pattern memory was normal in a natural variant rover (for(R)) but was impaired in another natural variant sitter (for(S)), which has a lower PKG level. Memory defects in for(S) flies could be rescued by either constitutive or adult-limited expression of for in the fan-shaped body. Interestingly, we showed that such rescue also occurred when for was expressed in the ellipsoid body. Additionally, expression of for in the fifth layer of the fan-shaped body restored sufficient memory for the pattern parameter "elevation" but not for "contour orientation," whereas expression of for in the ellipsoid body restored sufficient memory for both parameters. Our study defines a Drosophila model for further understanding the role of cGMP-PKG signaling in associative learning/memory and the neural circuit underlying this for-dependent visual pattern memory.