In computational neuroscience, synaptic plasticity rules are often formulated in terms of firing rates. The predominant description of in vivo neuronal activity, however, is the instantaneous rate (or spiking probability). In this article we resolve this discrepancy by showing that fluctuations of the membrane potential carry enough information to permit a precise estimate of the instantaneous rate in balanced networks. As a consequence, we find that rate based plasticity rules are not restricted to neuronal activity that is stable for hundreds of milliseconds to seconds, but can be carried over to situations in which it changes every few milliseconds. We illustrate this, by showing that a voltage-dependent realization of the classical BCM rule achieves input selectivity, even if stimulus duration is reduced to a few milliseconds each.

A classic result of Erdős and Pósa states that any graph either contains k vertexdisjoint cycles or can be made acyclic by deleting at most O(k log k) vertices. Birmelé, Bondy, and Reed (2007) raised the following more general question: given numbers l and k, what is the optimal function f (l, k) such that every graph G either contains k vertex-disjoint cycles of length at least l or contains a set X of f (l, k) vertices that meets all cycles of length at least l? In this paper, we answer that question by proving that f (l, k) = Θ(kl + k log k). As a corollary, the tree-width of any graph G that does not contain k vertexdisjoint cycles of length at least l is of order O(kl + k log k). This is also optimal up to constant factors and answers another question of Birmelé, Bondy, and Reed (2007).

Sequences of precisely timed neuronal activity are observed in many brain areas in various species. Synfire chains are a well-established model that can explain such sequences. However, it is unknown under which conditions synfire chains can develop in initially unstructured networks by self-organization. This work shows that with spike-timing dependent plasticity (STDP), modulated by global population activity, long synfire chains emerge in sparse random networks. The learning rule fosters neurons to participate multiple times in the chain or in multiple chains. Such reuse of neurons has been experimentally observed and is necessary for high capacity. Sparse networks prevent the chains from being short and cyclic and show that the formation of specific synapses is not essential for chain formation. Analysis of the learning rule in a simple network of binary threshold neurons reveals the asymptotically optimal length of the emerging chains. The theoretical results generalize to simulated networks of conductance-based leaky integrate-and-fire (LIF) neurons. As an application of the emerged chain, we propose a one-shot memory for sequences of precisely timed neuronal activity.

Fast bidirectional replays of place cell activity reflecting previous paths, and stripped off any instantial specifics of the animal's locomotion such as its speed or the duration of stops, have been observed during rest in rodents. Mechanisms underlying replays are not fully understood, as previous models depend on assumptions about the path, and on instantial specifics of motion. Relying on sharp-wave events, dendritic spikes and cholinergic modulation, we propose a spiking network model that stores traversed paths on a behavioral timescale with single exposure and produces fast bidirectional replays of corresponding place cell sequences independent of instantial specifics and the path taken. With the model, we make an experimentally verifiable prediction, the sequence cell population, whose firing follows a predefined sequential activity pattern independent of the environment. Furthermore, we hypothesize a functional role for disinhibition as behavioral time pacemaker, enforcing progression of sequence cell activity to match place sequences traversed.

A classic result of Erdős and Pósa says that any graph contains either k vertexdisjoint cycles or can be made acyclic by deleting at most O(k log k) vertices. Here we generalize this result by showing that for all numbers k and l and for every graph G, either G contains k vertex-disjoint cycles of length at least l, or there exists a set X of O(kl + k log k) vertices that meets all cycles of length at least l in G. As a corollary, the tree-width of any graph G that does not contain k vertex-disjoint cycles of length at least l is of order O(kl + k log k). These results improve on the work of Birmelé, Bondy and Reed '07 and Fiorini and Herinckx '14 and are optimal up to constant factors.