Numbers and brains.

The representation of discrete and continuous quantities appears to be ancient and pervasive in animal brains. Because numbers are the natural carriers of these representations, we may discover that in brains, it's numbers all the way down.

Bayesian change-point analysis reveals developmental change in a classic theory of mind task.

Although learning and development reflect changes situated in an individual brain, most discussions of behavioral change are based on the evidence of group averages. Our reliance on group-averaged data creates a dilemma. On the one hand, we need to use traditional inferential statistics. On the other hand, group averages are highly ambiguous when we need to understand change in the individual; the average pattern of change may characterize all, some, or none of the individuals in the group. Here we present a new method for statistically characterizing developmental change in each individual child we study. Using false-belief tasks, fifty-two children in two cohorts were repeatedly tested for varying lengths of time between 3 and 5 years of age. Using a novel Bayesian change point analysis, we determined both the presence and-just as importantly-the absence of change in individual longitudinal cumulative records. Whenever the analysis supports a change conclusion, it identifies in that child's record the most likely point at which change occurred. Results show striking variability in patterns of change and stability across individual children. We then group the individuals by their various patterns of change or no change. The resulting patterns provide scarce support for sudden changes in competence and shed new light on the concepts of "passing" and "failing" in developmental studies.

Mice take calculated risks.

Animals successfully navigate the world despite having only incomplete information about behaviorally important contingencies. It is an open question to what degree this behavior is driven by estimates of stochastic parameters (brain-constructed models of the experienced world) and to what degree it is directed by reinforcement-driven processes that optimize behavior in the limit without estimating stochastic parameters (model-free adaptation processes, such as associative learning). We find that mice adjust their behavior in response to a change in probability more quickly and abruptly than can be explained by differential reinforcement. Our results imply that mice represent probabilities and perform calculations over them to optimize their behavior, even when the optimization produces negligible material gain.

The neuroscience of learning: beyond the Hebbian synapse.

From the traditional perspective of associative learning theory, the hypothesis linking modifications of synaptic transmission to learning and memory is plausible. It is less so from an information-processing perspective, in which learning is mediated by computations that make implicit commitments to physical and mathematical principles governing the domains where domain-specific cognitive mechanisms operate. We compare the properties of associative learning and memory to the properties of long-term potentiation, concluding that the properties of the latter do not explain the fundamental properties of the former. We briefly review the neuroscience of reinforcement learning, emphasizing the representational implications of the neuroscientific findings. We then review more extensively findings that confirm the existence of complex computations in three information-processing domains: probabilistic inference, the representation of uncertainty, and the representation of space. We argue for a change in the conceptual framework within which neuroscientists approach the study of learning mechanisms in the brain.

The physical basis of memory.

Neuroscientists are searching for the engram within the conceptual framework established by John Locke's theory of mind. This framework was elaborated before the development of information theory, before the development of information processing machines and the science of computation, before the discovery that molecules carry hereditary information, before the discovery of the codon code and the molecular machinery for editing the messages written in this code and translating it into transcription factors that mark abstract features of organic structure such as anterior and distal. The search for the engram needs to abandon Locke's conceptual framework and work within a framework informed by these developments. The engram is the medium by which information extracted from past experience is transmitted to the computations that inform future behavior. The information-conveying symbols in the engram are rapidly generated in the course of computations, which implies that they are molecules.

Robert A. Rescorla (1940-2020).

Memorializes Robert Arthur Rescorla (1940-2020), emeritus professor of psychology at the University of Pennsylvania. Rescorla was the world's most distinguished scholar in animal learning and a great teacher. Rescorla thought of himself as primarily an experimen talist, and his experiments on Pavlovian conditioning and instrumental learning would win any prize for the aesthetics of experimental design. Most of his nearly 200 empirical articles included multiple experiments with replications. Among Re scorla's most beautiful experiments were the studies of extinc tion performed in the last decade of his career. (PsycInfo Database Record (c) 2021 APA, all rights reserved).

Number and time in acquisition, extinction and recovery.

We measured rate of acquisition, trials to extinction, cumulative responses in extinction, and the spontaneous recovery of anticipatory hopper poking in a Pavlovian protocol with mouse subjects. We varied by factors of 4 number of sessions, trials per session, intersession interval, and span of training (number of days over which training extended). We find that different variables affect each measure: Rate of acquisition [1/(trials to acquisition)] is faster when there are fewer trials per session. Terminal rate of responding is faster when there are more total training trials. Trials to extinction and amount of responding during extinction are unaffected by these variables. The number of training trials has no effect on recovery in a 4-trial probe session 21 days after extinction. However, recovery is greater when the span of training is greater, regardless of how many sessions there are within that span. Our results and those of others suggest that the numbers and durations and spacings of longer-duration "episodes" in a conditioning protocol (sessions and the spans in days of training and extinction) are important variables and that different variables affect different aspects of subjects' behavior. We discuss the theoretical and clinical implications of these and related findings and conclusions-for theories of conditioning and for neuroscience.

The importance of proving the null.

Null hypotheses are simple, precise, and theoretically important. Conventional statistical analysis cannot support them; Bayesian analysis can. The challenge in a Bayesian analysis is to formulate a suitably vague alternative, because the vaguer the alternative is (the more it spreads out the unit mass of prior probability), the more the null is favored. A general solution is a sensitivity analysis: Compute the odds for or against the null as a function of the limit(s) on the vagueness of the alternative. If the odds on the null approach 1 from above as the hypothesized maximum size of the possible effect approaches 0, then the data favor the null over any vaguer alternative to it. The simple computations and the intuitive graphic representation of the analysis are illustrated by the analysis of diverse examples from the current literature. They pose 3 common experimental questions: (a) Are 2 means the same? (b) Is performance at chance? (c) Are factors additive?

The generative basis of natural number concepts.

Number concepts must support arithmetic inference. Using this principle, it can be argued that the integer concept of exactly ONE is a necessary part of the psychological foundations of number, as is the notion of the exact equality - that is, perfect substitutability. The inability to support reasoning involving exact equality is a shortcoming in current theories about the development of numerical reasoning. A simple innate basis for the natural number concepts can be proposed that embodies the arithmetic principle, supports exact equality and also enables computational compatibility with real- or rational-valued mental magnitudes.

Neuron function inferred from behavioral and electrophysiological extimates of refractory period.

The refrcactory period of neurons mediating an electrically elicited behavior (self-stimulation) was estimated by varying the intrapair pulse separation in a stimulating train made up of pulse pairs and measuring the intensity of the elicited behavior. Two neuronal systems with different refractory periods were indicated. Single-unit recording in acute preparations stimulated through self-stimulation electrodes revealed primarily two classes of units. Each class gave refractory period estimates characteristic of one of the behaviorally indicated systems. The experiments illustrate a technique for establishing functional relations between single units in the brain and gross behavior.

Contingency, contiguity, and causality in conditioning: Applying information theory and Weber's Law to the assignment of credit problem.

Contingency is a critical concept for theories of associative learning and the assignment of credit problem in reinforcement learning. Measuring and manipulating it has, however, been problematic. The information-theoretic definition of contingency-normalized mutual information-makes it a readily computed property of the relation between reinforcing events, the stimuli that predict them and the responses that produce them. When necessary, the dynamic range of the required temporal representation divided by the Weber fraction gives a psychologically realistic plug-in estimates of the entropies. There is no measurable prospective contingency between a peck and reinforcement when pigeons peck on a variable interval schedule of reinforcement. There is, however, a perfect retrospective contingency between reinforcement and the immediately preceding peck. Degrading the retrospective contingency by gratis reinforcement reveals a critical value (.25), below which performance declines rapidly. Contingency is time scale invariant, whereas the perception of proximate causality depends-we assume-on there being a short, fixed psychologically negligible critical interval between cause and effect. Increasing the interval between a response and reinforcement that it triggers degrades the retrograde contingency, leading to a decline in performance that restores it to at or above its critical value. Thus, there is no critical interval in the retrospective effect of reinforcement. We conclude with a short review of the broad explanatory scope of information-theoretic contingencies when regarded as causal variables in conditioning. We suggest that the computation of contingencies may supplant the computation of the sum of all future rewards in models of reinforcement learning. (PsycINFO Database Record (c) 2019 APA, all rights reserved).

Time has come.

The ramp-like rise and fall of activity in neurons of the LIP area of the posterior parietal cortex of alert behaving monkeys performing a duration discrimination task tracks the changing relative likelihoods that the stimulus in their response field will become the target of a saccade.

Intact interval timing in circadian CLOCK mutants.

While progress has been made in determining the molecular basis for the circadian clock, the mechanism by which mammalian brains time intervals measured in seconds to minutes remains a mystery. An obvious question is whether the interval-timing mechanism shares molecular machinery with the circadian timing mechanism. In the current study, we trained circadian CLOCK +/- and -/- mutant male mice in a peak-interval procedure with 10 and 20-s criteria. The mutant mice were more active than their wild-type littermates, but there were no reliable deficits in the accuracy or precision of their timing as compared with wild-type littermates. This suggests that expression of the CLOCK protein is not necessary for normal interval timing.

Flawed foundations of associationism? Comments on Machado and Silva (2007).

A. Machado and F. J. Silva have spotted an important conceptual problem in scalar expectancy theory's account of the 2-standard-interval time-left experiment. C. R. Gallistel and J. Gibbon (2000) were aware of it but did not discuss it for historical and sociological reasons, owned up to in this article. A problem of broader significance for psychology, cognitive science, neuroscience, and the philosophy of mind concerns the closely related concepts of a trial and of temporal pairing, which are foundational in associative theories of learning and memory. Association formation is assumed to depend on the temporal pairing of the to-be-associated events. In modeling it, theorists have assumed continuous time to be decomposable into trials. But life is not composed of trials, and attempts to specify the conditions under which two events may be regarded as temporally paired have never succeeded. Thus, associative theories of learning and memory are built on conceptual sand. Undeterred, neuroscientists have defined the neurobiology-of-memory problem as the problem of determining the cellular and molecular mechanism of association formation, and connectionist modelers have made it a cornerstone of their efforts. More conceptual analysis is indeed needed.

Time left in the mouse.

Evidence suggests that the online combination of non-verbal magnitudes (durations, numerosities) is central to learning in both human and non-human animals [Gallistel, C.R., 1990. The Organization of Learning. MIT Press, Cambridge, MA]. The molecular basis of these computations, however, is an open question at this point. The current study provides the first direct test of temporal subtraction in a species in which the genetic code is available. In two experiments, mice were run in an adaptation of Gibbon and Church's [Gibbon, J., Church, R.M., 1981. Time left: linear versus logarithmic subjective time. J. Exp. Anal. Behav. 7, 87-107] time left paradigm in order to characterize typical responding in this task. Both experiments suggest that mice engaged in online subtraction of temporal values, although the generalization of a learned response rule to novel stimulus values resulted in slightly less systematic responding. Potential explanations for this pattern of results are discussed.

Is matching innate?

Experimentally naive mice matched the proportions of their temporal investments (visit durations) in two feeding hoppers to the proportions of the food income (pellets per unit session time) derived from them in three experiments that varied the coupling between the behavioral investment and food income, from no coupling to strict coupling. Matching was observed from the outset; it did not improve with training. When the numbers of pellets received were proportional to time invested, investment was unstable, swinging abruptly from sustained, almost complete investment in one hopper, to sustained, almost complete investment in the other-in the absence of appropriate local fluctuations in returns (pellets obtained per time invested). The abruptness of the swings strongly constrains possible models. We suggest that matching reflects an innate (unconditioned) program that matches the ratio of expected visit durations to the ratio between the current estimates of expected incomes. A model that processes the income stream looking for changes in the income and generates discontinuous income estimates when a change is detected is shown to account for salient features of the data.

The Coding Question.

Recent electrophysiological results imply that the duration of the stimulus onset asynchrony in eyeblink conditioning is encoded by a mechanism intrinsic to the cerebellar Purkinje cell. This raises the general question - how is quantitative information (durations, distances, rates, probabilities, amounts, etc.) transmitted by spike trains and encoded into engrams? The usual assumption is that information is transmitted by firing rates. However, rate codes are energetically inefficient and computationally awkward. A combinatorial code is more plausible. If the engram consists of altered synaptic conductances (the usual assumption), then we must ask how numbers may be written to synapses. It is much easier to formulate a coding hypothesis if the engram is realized by a cell-intrinsic molecular mechanism.

Finding numbers in the brain.

After listing functional constraints on what numbers in the brain must do, I sketch the two's complement fixed-point representation of numbers because it has stood the test of time and because it illustrates the non-obvious ways in which an effective coding scheme may operate. I briefly consider its neurobiological implementation. It is easier to imagine its implementation at the cell-intrinsic molecular level, with thermodynamically stable, volumetrically minimal polynucleotides encoding the remembered numbers, than at the circuit level, with plastic synapses encoding them.This article is part of a discussion meeting issue 'The origins of numerical abilities'.

Navigation: Whence Our Sense of Direction?

Behavioral data have long implied our sense of direction derives from global environmental shape; electrophysiological evidence, however, has seemed to imply it derives from salient non-geometric landmarks. Experiments on the re-establishment of place fields in disoriented mice now align the electrophysiological data with the behavioral data.

Theoretical implications of quantitative properties of interval timing and probability estimation in mouse and rat.

In three experiments with mice ( Mus musculus ) and rats (Rattus norvigicus), we used a switch paradigm to measure quantitative properties of the interval-timing mechanism. We found that: 1) Rodents adjusted the precision of their timed switches in response to changes in the interval between the short and long feed latencies (the temporal goalposts). 2) The variability in the timing of the switch response was reduced or unchanged in the face of large trial-to-trial random variability in the short and long feed latencies. 3) The adjustment in the distribution of switch latencies in response to changes in the relative frequency of short and long trials was sensitive to the asymmetry in the Kullback-Leibler divergence. The three results suggest that durations are represented with adjustable precision, that they are timed by multiple timers, and that there is a trial-by-trial (episodic) record of feed latencies in memory.