Plasticity is the capability of neurons and synapses to change their parameters depending on their past activity. Most frequently, we speak of synaptic plasticity, meaning the activity-dependent change of the synaptic weights, but we can con- sider changes in other neuron parameters as plasticity as well. Activity-dependent plasticity is generally considered as the basic phenomenon behind the learning of motoric skills and formation of memories [Dayan and Abbott, 2001]. A histori- cal review on plasticity and on the perspectives of plasticity research is given in Markram et al. [2011a].
2.2 Computational neuroscience: neurons, synapses and plasticity There are several models trying to explain experimental evidences regarding synaptic plasticity. Each of them relies on or explains aspects of plasticity ex- periments but the true plasticity rule(s) of the brain remain(s) elusive. The first qualitative description of synaptic plasticity goes back to Hebb’s famous rule, often summarized as: if neurons fire together, they wire together [Hebb, 1949]. How- ever, Hebb did not gave a mathematical formula to his qualitative description.7 The most common requirement for plasticity is locality. Generally, we assume that the equations governing the change of the neuron parameters and synaptic weights should only depend on locally accessible variables, such as the spiking activity of the pre- and post-synaptic neurons. This requirement is often violated by implicitly requiring non-local computations for plasticity, for example Zenke and Ganguli [2018].
Note, that there is a difference between local variables and locally accessible variables. Locally accessible variables are quantities that are plausible accessible by the neurons and synapses to influence their plasticity. Local variables are specific for any given neuron or synapse, for example a function of the pre- and post-synaptic activities. By a global variable we mean a quantity that is the same over the entire (or at least large parts of) the neural network. An example for a global variable are the neuromodulators: neurotransmitters that not only influence the post-synaptic neuron but a larger group of neurons by, for instance, modifying their plasticity, see for example Gerstner et al. [2018]. In this sense, neuromodulators are global variables that are at the same time locally accessible. In the following, we give the basic and simple structure of formulating plasticity rules proposed by Gerstner and Kistler [2002a]. Rate-based plasticity can be written in the general form:
dwij
dt = Frate(νi, νj; wij) , (2.21)
where Frateis a general function of the post- and pre-synaptic firing rates (activi-
ties) νiand νjas well as the synaptic weight wij. By taking the Taylor expansion
of Frateup to a given degree, we can reconstruct many rate-based plasticity rules
found in literature, for example the plasticity models in Oja [1982] and Bienenstock et al. [1982]. This formulation only implicitly includes plasticity rules defined on neuron models with spatial extension and plasticity rules using membrane potentials. Still, it captures the main belief of locality in synaptic rules.
Similarly, for spike-based models, we write: dwij dt = Fspike(S post i (t ′) , Sprej (t”); wij) , (2.22)
where Sposti (t′)and Sprej (t”)are the spike-trains from the post- and pre-synaptic neurons. Fspikeis now a functional of the spike-trains. The different times t′and
7Hebb’s original formulation was: “When an axon of cell A is near enough to excite a cell B
and repeatedly or persistently takes part in firing it, some growth process or metabolic change takes place in one or both cells such that A’s efficiency, as one of the cells firing B, is increased.” — from Hebb [1949].
2. Background
t” indicate that the rate of change of the synaptic weight does not only depend on the current spiking properties but can also shows memory effects. A Volterra expansion of Fspikeyields the different spike-based plasticity models.8 For example,
the classic spike-time-dependent plasticity (STDP) learning rule — first observed experimentally by Bi and Poo [1998] — can be written as:
dwij dt = ∫ t −∞W(t ′− t)Sposti (t′)Sprej (t)dt′
post before pre
+ ∫ t −∞W(t−t ′) Sposti (t)Sprej (t′)dt′
pre before post
,
(2.23) where W(tpre−tpost)is the STDP function. The choice of the variables tpreand tpost
highlights that the STDP function always takes relative spike-timing differences as an argument. A popular choice for the STDP function is the exponential kernel (see also figure 2.9):
W(tpre−tpost) = ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ a+exp ( −tpost−tpre τ+ ) if tpre<tpost a−exp ( −tpre−tpost τ− ) if tpre>tpost , (2.24)
where a+and τ+parameterizes the causal (pre-before-post) branch and a−and τ−
parameterizes the anti-causal (post-before-pre) branch of the spike-time-dependent plasticity (STDP) function. Equation (2.23) is formulated to evoke synaptic plastic- ity over all spike pairing, but often STDP is formulated in a way to only include next neighbor spike-pairs. After the discovery of Bi and Poo [1998], STDP was considered as always causally strengthening and anti-causally weakening. Nowa- days, as it is also apparent from the general formulation of the theory, the STDP function is not looked at as a hard-wired rule. Instead, it is thought of as a building block for more complex learning rules based on causal and anti-causal correlation measurements, which are carried out locally at the synapses.