Firing statistics of inhibitory neuron with delayed feedback. I. Output ISI probability density

Activity of inhibitory neuron with delayed feedback is considered in the framework of point stochastic processes. The neuron receives excitatory input impulses from a Poisson stream, and inhibitory impulses from the feedback line with a delay. We investigate here, how does the presence of inhibitory feedback affect the output firing statistics. Using binding neuron (BN) as a model, we derive analytically the exact expressions for the output interspike intervals (ISI) probability density, mean output ISI and coefficient of variation as functions of model's parameters for the case of threshold 2. Using the leaky integrate-and-fire (LIF) model, as well as the BN model with higher thresholds, these statistical quantities are found numerically. In contrast to the previously studied situation of no feedback, the ISI probability densities found here both for BN and LIF neuron become bimodal and have discontinuity of jump type. Nevertheless, the presence of inhibitory delayed feedback was not found to affect substantially the output ISI coefficient of variation. The ISI coefficient of variation found ranges between 0.5 and 1. It is concluded that introduction of delayed inhibitory feedback can radically change neuronal output firing statistics. This statistics is as well distinct from what was found previously (Vidybida & Kravchuk, 2009) by a similar method for excitatory neuron with delayed feedback.Comment: 23 pages, 8 figure

Endogeneous price leadership in a duopoly: equal products, unequal technology

In the present paper we study endogenous price leadership in the context of a homogeneous product Bertrand duopoly model in which the firms have different, strictly convex cost functions. In such a framework it is well known that a simultaneous move price choice game does not have an equilibrium in pure strategies, but it has an equilibrium in mixed strategies. In the Stackelberg games with an exogenous price leader, we show that a pure strategy subgame perfect Nash equilibrium (SPNE) always exists. Although the SPNE might not be unique, the payoffs are the same across all SPNE. Finally, we analyze the issue of endogenous price leadership using the continuous version of the Robson (1990) timing game. The result is unexpected. One would expect the more efficient firm to emerge as the endogenous price leader. This is not always true. In most cases the endogenous leader is the firm with the highest "threshold" price. However, we also provide conditions under which the more efficient firm emerges as the leader. Our paper essentially complements Yano (2001), which is based on the Hamilton and Slutsky (1990) framework

Bertrand Games and Sharing Rules

Bertrand games, Sharing rule, Tie-decreasing sharing rule, Coalition monotonicity, C72, D43, L13,