Dynamics and Learning

This chapter explores questions related to learning and dynamics. The first part explores dynamic quantal response equilibrium models where strategies are conditioned on observed histories of past decisions and outcomes of stage games. The second part considers models in which players are learning about others' behavior via a process in which they may update and respond to current beliefs in a noisy (quantal) manner. The final section explores learning models that involve quantal responses to beliefs formed by processing information from finite (but possibly long) histories of prior or observed action profiles. The formulation permits consideration of a wide variety of exogenous or even endogenous (e.g., least squares) learning rules.

Properties of Quantal Transmission at CA1 Synapses

We have used Monte Carlo simulations to understand the generation of quantal responses at the single active zones of CA1 synapses. We constructed a model of AMPA channel activation that accounts for the responses to controlled glutamate application and a model of glutamate diffusion in the synaptic cleft. With no further adjustments to these models, we simulated the response to the release of glutamate from a single vesicle. The predicted response closely matches the rise time of observed responses, which recent measurements show is much faster (<100 μs) than previously thought. The simulations show that initial channel opening is driven by a brief (<100 μs) glutamate spike near the site of vesicle fusion, producing a hotspot of channel activation (diameter: ∼250 nm) smaller than many synapses. Quantal size therefore depends more strongly on the density of channels than their number, a finding that has important implications for measuring synaptic strength. Recent measurements allow estimation of AMPA receptor density at CA1 synapses. Using this value, our simulations correctly predicts a quantal amplitude of ∼10 pA. We have also analyzed the properties of excitatory postsynaptic currents (EPSCs) generated by the multivesicular release that can occur during evoked responses. We find that summation is nearly linear and that the existence of multiple narrow peaks in amplitude histograms can be accounted for. It has been unclear how to reconcile the existence of these narrow peaks, which indicate that the variation of quantal amplitude is small (CV < 0.2) with the highly variable amplitude of miniature EPSCs (mEPSCs; CV ∼ 0.6). According to one theory, mEPSC variability arises from variation in vesicle glutamate content. However, both our modeling results and recent experimental results indicate that this view cannot account for the observed rise time/amplitude correlation of mEPSCs. In contrast, this correlation and the high mEPSC variability can be accounted for if some mEPSCs are generated by two or more vesicles released with small temporal jitter. We conclude that a broad range of results can be accounted for by simple principles: quantal amplitude (∼10 pA) is stereotyped, some mEPSCs are multivesicular at moderate and large synapses, and evoked responses are generated by quasi-linear summation of multiple quanta.

Kinetic model of the inositol trisphosphate receptor that shows both steady-state and quantal patterns of Ca2+ release from intracellular stores

The release of Ca2+ from intracellular stores via InsP3 receptors shows anomalous kinetics. Successive additions of low concentrations of InsP3 cause successive rapid transients of Ca2+ release. These quantal responses have been ascribed to all-or-none release from stores with differing sensitivities to InsP3 or, alternatively, to a steady-state mechanism where complex kinetic properties of the InsP3 receptor allow partial emptying of all the stores. We present here an adaptive model of the InsP3 receptor that can show either pattern, depending on the imposed experimental conditions. The model proposes two interconvertible conformational states of the receptor: one state binds InsP3 rapidly, but with low affinity, whereas the other state binds slowly, but with high affinity. The model shows repetitive increments of Ca2+ release in the absence of a Ca2+ gradient, but more pronounced incremental behaviour when released Ca2+ builds up at the mouth of the channel. The sensitivity to InsP3 is critically dependent on the density of InsP3 receptors, so that different stores can respond to different concentration ranges of InsP3. Since the model generates very high Hill coefficients (h7), it allows all-or-none release of Ca2+ from stores of differing receptor density, but questions the validity of the use of h values as a guide to the number of InsP3 molecules needed to open the channel. The model presents a mechanism for terminating Ca2+ release in the presence of positive feedback from released Ca2+, thereby providing an explanation of why elementary Ca2+ signals ('blips’ and ‘puffs') do not inevitably turn into regenerative waves.