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Until now, my research in psychology has been limited to behavioral theories. I’m not a radical behaviorist, but one has to start somewhere, and behaviorism was, for several reasons, the most natural choice for someone coming from physics like me. And, finally, Yes, this liberating feeling of being able to test my ideas directly, with my own hands, in an experiment. If you read about my research career before taking the red pill, you understand what I mean. The red pill is fun.

The geometry of learning

I have applied the mathematical tools used in physics to models of animal and human behavior. In a recent paper, I have reformulated classic models of Pavlovian conditioning such as Hull’s, Rescorla-Wagner’s and Mackintosh’s with the elegant language of fractal geometry. The advantage in doing so is that one can define quantities such as the Hausdorff dimension of a fractal and express them in terms of the parameters of the set, which, in this case, are the saliences of the conditioned (CS) and unconditioned stimuli (US) and the probability of presentation of the US. In this way, one can catalogue different learning processes according to the dimensionality of their associated fractal set.

The association strength at a given training trial corresponds to a point in a disconnected set at a given iteration level. In this way, one can represent a training process as a hopping on a fractal set, instead of the traditional learning curve as a function of the trial. The main advantage of this novel perspective is to provide an elegant classification of associative theories in terms of the geometric features of fractal sets. In particular, the dimension of fractals is a parameter that can both measure the efficiency of a given conditioning model (in terms of the salience of the stimuli for the experimental subject) and compare the efficiency of different models.

I illustrated the correspondence with the examples of the Hull, Rescorla-Wagner, and Mackintosh models and show that they are equivalent to a Cantor set. More generally, conditioning programs are described by the geometry of their associated fractal (typically random), which gives much more information than just its dimension.

A random ternary Cantor set corresponding to a randomized Hull model of the first type for n=7 iterations. The thin red (blue) solid curve is the learning curve of excitatory conditioning (respectively, extinction) in a partial reinforcement program. Dashed curves are the learning curves in the deterministic model. Source: arXiv:1605.00591.

There also is a possible relation between this formalism and other “fractal” findings in the cognitive literature, in particular the famous 1/f noise in human performance. The Hausdorff dimension of the fractals associated with conditioning processes decreases (from 1 to 0) with the efficiency of conditioning, while the Hausdorff or “fractal” dimension of the stochastic response pattern in cognitive tasks decreases (to values above or near 2) with the improvement of the performance. In the first case, a better performance means a faster conditioning and a more dust-like set, while in the second case a better performance literally means a smoother performance. The quantitative theoretical description I proposed yields cautious support to what found experimentally in cognitive science and to the notion that measurable behaviour can be characterized, in a precise sense, by irregular geometry.

Response variability: theoretical models

This quantitative theory goes beyond classical associative models. These models, such as Hull’s, can be described as the dynamics of a point particle rolling on a potential, where the particle “position” is the association strength established between the CS and the US. The novelty we propose is not so much this description in terms of dynamics, but its quantization. This technical step leads to new predictions. Once the subject has reached the asymptote of learning, instead of sitting there forever as in classical models its response fluctuates according to a precise mathematical model. skinnerboxFrom January 2017 to January 2018, I conducted an experiment at the Laboratory of Animal Behavior of the Department of Basic Psychology of UNED University (Madrid) where, in collaboration with Ricardo Pellón and Ernesto Caballero, we tested this phenomenon.