Spike timing dependent plasticity (STDP) is one of the simplest yet key mechanisms enabling functional adaptation in neuronal systems (see e.g. 
and references therein). Broadly speaking, if we consider two synaptically connected cells, STDP stands for a change in synaptic efficacy as a function of timing between pre- and post- synaptic events. If the pos-synaptic event occurs within a given interval of time from the onset of the pre-synaptic one then efficacy of synaptic transmission enhances. If, however, the opposite takes place, i.e. a post-synaptic event is followed by pre-synaptic spike, then the efficacy decreases. Despite overall apparent simplicity of the phenomenon, it allows to link higher cognitive functions such as learning and memory with molecular and cellular processes underlying signal transmission in neuronal networks. Various interesting aspects of STDP in relation to bidirectional plasticity and bistability have been discussed and analyzed in the literature 
. In addition, as it has been shown in 
, STDP may be involved in the formation of metaplasticity 
. With respect to the function, STDP is a component of plausible models of selective attention 
and working memory 
. At the lower scale of functional organization, STDP may trigger long-term potentiation (LTP) or depression (LTP) 
. Finally, STDP is believed to play a role in phase coding – a way of representing information about stimuli in terms of the relative time moments of spike occurrences.
Many forms of STDP have been discovered to date 
, and a common knowledge is that STDP is supported by multiple molecular cascades inducing changes in both postsynaptic spines and in presynaptic terminals. Calcium flux through NMDA receptors located in spines 
is an example of mechanisms directly responsible for postsynaptic changes. In this mechanism, excitatory postsynaptic potentials preceding back-propagating action potentials elicit calcium influx through postsynaptic NMDA receptors. Higher calcium concentration, in turn, facilitates evoking of postsynaptic spikes in response to the presynaptic ones. Changes in presynaptic terminals are observed, for example, in the hippocampal mossy fiber synapses 
. STDP-like phenomena can also occur due to the modulation of synaptic transmission by endocannabinoid-mediated retrograde cascades. These cascades, once activated, trigger the activation of presynaptic receptors 
Large diversity of the ways in which STDP may manifest itself in empirical observations has lead to a broad range of mathematical models of the phenomenon. These models, although phenomenological, are widely used in computational and theoretical studies (see e.g. 
). In the majority of these models the principal factor determining synaptic efficacy is the synaptic weight. The latter is described by a dynamic variable of which the value changes in response to post-to-presynaptic spike timing. Increments/decrements of the weights are often associated to LTP/LTD respectively. One of the outcomes of such activity-dependent modifications of the synaptic weights is that connections between individual cells may grow or decay over time by a relatively large amount. This facilitates emergence of neuronal clusters that fire together, up to a tolerance margin.
A particular form of such firing activity in which clusters of neurons produce time-locked spiking sequences
has recently received substantial attention in the literature 
. Relative time lags between spikes in these sequences are robust; the sequences can repeat spontaneously, or they can be generated in response to a certain stimulus. A number of theoretical frameworks have been proposed to explain emergence and persistence of these precise firing patterns with different inter-spike timing, see e.g. 
and related notions of synchronized chains (synfire chains) and polychronous groups. In these frameworks STDP, linked to the post-to-presynaptic timing, is advocated as a mechanism that is directly responsible for the emergence of persistent spike sequences within a given topological substrate. Even though computational evidence suggests that this may indeed be the case, rigorous correspondence between stimuli, particular STDP-based signaling pathways, and their stability is not yet fully understood. In particular, the question of how STDP may ensure precise timing of spiking sequences with arbitrary lags between spikes is still open. Finding an answer to this question is the main goal of our current work.
In this paper we investigate dynamic properties of a pair of neural oscillators coupled via synaptic STDP-enabled connections. Our results suggest that for this class of systems accurate tuning of post-to-presynaptic spike timing to a given, and broadly arbitrary, value is indeed possible via a suitable STDP mechanism. This mechanism can be viewed as a feedback facilitating or depressing synaptic transmission “on demand”, depending on timing of stimulation. In contrast to conventional models of STDP in which spike-timing modulates weights of synaptic connections, we consider a model of STDP in which spike-timing influences internal state of pre- or post- synaptic neurons. Such internal state is, in the case of our model, an excitation parameter enhancing/suppressing spike generation. This feature of spike-dependent potentiation is well-documented phenomenologically 
. We show that coarse tuning of spike timing is readily achievable in a pair of interconnected neural oscillators equipped with such STDP mechanism. Further fine-tuning of spiking patterns can be achieved via additional slow fluctuations of the base line of excitation thresholds.
The main motivation for choosing excitation-driven STDP mechanisms rather than conventional models of STDP (i.e. the ones modulating the weights of connections) is that we would like to be able to deal with realistic cases of neurons having different natural frequencies. As a general rule, the larger the difference between natural frequencies of neural oscillations the larger should be the values of synaptic weights if accurate time-locking of spikes is desired, cf. e.g. 
. This, however, may conflict with the standard assumption demanding that coupling between elements in the system is weak. Thus regulatory mechanisms complementary to the ones modulating the values of synaptic weights are needed for ensuring precise locking of spike sequences in systems of neurons with inherently non-identical frequencies of spike generation. STDP-driven modification of excitation variables is a plausible candidate for this role.
For the sake of numerical and analytical tractability we focus predominantly on a simplified spike transmission model using a pair of neuronal oscillators coupled via excitatory synaptic coupling. Synaptic transmission in the model is unidirectional and instantaneous: a spike in the postsynaptic neuron is evoked as soon as the excitatory postsynaptic potential (EPSP) exceeds certain threshold. As a model for pre- and post-synaptic neurons we use Rowat-Selverston neuronal oscillator 
. This model is computationally efficient, yet being a reduction of Hodgkin-Huxley classical model, it bears a fair degree of biological realism. The model is typically used in computational studies of synchronization and phase-locking in networks of synaptically coupled cells 
. Here we also employ this model for studying phase-locking behavior of neurons with STDP-enabled synaptic connections.
The paper is organized as follows. Section Methods
contains description of the Rowat-Selverston neuronal oscillator and also specifies the class of synaptic coupling considered in the paper. In addition, it presents the concept of phase spiking maps which is used in both numerical and analytical parts of the study. Definitions of specific STDP mechanisms are provided in Results
. This is followed by quantitative and qualitative description of the dynamics such mechanisms may induce in the coupled system. The results are summarized and discussed in a brief Discussion
. Technical derivations and other auxiliary materials are presented in Appendix S1