In neuroscience, learning and memory are usually associated to long-term changes of connection strength between neurons . In this context, synaptic plasticity refers to the set of mechanisms driving the dynamics of neuronal connections, called synapses {\em  and represented by a scalar value, the synaptic weight. A Spike-Timing Dependent Plasticity (STDP) rule is a biologically-based model representing the time evolution of the synaptic weight as a functional of the past spiking activity of adjacent neurons. If numerous models of neuronal cells have been proposed in the mathematical literature, few of them include a variable for the time-varying strength of the connection. In this article, a new, general, mathematical framework to study the phenomenon of synaptic plasticity associated to STDP rules is introduced. A system composed of two neuronal cells connected by a single synapse is investigated and a stochastic process describing its dynamical behavior is presented and analyzed. The notion of plasticity kernel is introduced as a key component of plastic neural networks models  . We show that a large number of STDP rules from neuroscience and physics applied to neural systems can be represented by this formalism. Experiments show that long-term synaptic plasticity evolves on a much slower timescale than the cellular mechanisms driving the activity of neuronal cells. For this reason a scaling model of our stochastic model is also introduced and averaging principles for a sub-class of plasticity kernels are stated, and proved in a companion paper. These results are used to analyze two STDP models widely used in applied physics: Pair-based rules and calcium-based rules. We compare results of computational neuroscience on models of timing-based synaptic plasticity with our results derived from averaging principles .
In neuroscience, learning and memory are usually associated to long-term changes of neuronal connectivity . In this context, synaptic plasticity refers to the set of mechanisms driving the dynamics of neuronal connections, called {\em synapses and represented by a scalar value, the synaptic weight. Spike-Timing Dependent Plasticity (STDP) is a biologically-based model representing the time evolution of the synaptic weight as a functional of the past spiking activity of adjacent neurons. If numerous models of neuronal cells have been proposed in the mathematical literature, few of them include a variable for the time-varying strength of the connection. A new, general, mathematical framework is introduced to study synaptic plasticity associated to different STDP rules. The system composed of two neurons connected by a single synapse is investigated and a stochastic process describing its dynamical behavior is presented and analyzed. The notion of plasticity kernel is introduced as a key component of plastic neural networks models , generalizing a notion used for pair-based models . We show that a large number of STDP rules from neuroscience and physics can be represented by this formalism. Several aspects of these models are discussed and compared to canonical models of computational neuroscience . An important sub-class of plasticity kernels with a Markovian formulation is also defined and investigated. In these models, the time evolution of cellular processes such as the neuronal membrane potential and the concentrations of chemical components created/suppressed by spiking activity has the Markov property .
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connection strength between neurons
neuronal connectivity
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synapses
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synapses
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D
rule
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In this article, a
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to study the phenomenon of
is introduced to study
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STDP rules is introduced. A
different STDP rules. The
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neuronal cells
neurons
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, generalizing a notion used for pair-based models
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applied to neural systems
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Experiments show that long-term synaptic plasticity evolves on a much slower timescale than the cellular mechanisms driving the activity of neuronal cells. For this reason a scaling model of our stochastic model is also introduced and averaging principles for a sub-class of plasticity kernels are stated, and proved in a companion paper. These results are used to analyze two STDP models widely used in applied physics: Pair-based rules and calcium-based rules. We compare results
Several aspects of these models are discussed and compared to canonical models
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on models of timing-based synaptic plasticity with our results derived from averaging principles
. An important sub-class of plasticity kernels with a Markovian formulation is also defined and investigated. In these models, the time evolution of cellular processes such as the neuronal membrane potential and the concentrations of chemical components created/suppressed by spiking activity has the Markov property