The Formation of Multi-synaptic Connections by the Interaction of Synaptic and Structural Plasticity and Their Functional Consequences
Figure 2
Possible distributions of the number of synapses of a single connection resulting from the interaction of synaptic and structural plasticity with different neuron models.
(A) Three different curvatures of input-output functions F of the neuron (black) lead to different shapes (curvatures) of the combinatorial term pcf (red, see Eq. 4). For fixed presynaptic activity and postsynaptic stimulation, the lines are calculated for continuous values of S, whereas the dots mark successive discrete values. (B) When the combinatorial influences pcf (red, Eq. 4) are smaller than the logarithmic deletion probability pd (black) for a certain value of S (grey shaded area), the long-term equilibrium probability for S synapses is higher than the probability for S − 1 synapses (see Eq. 2) and vice versa. Thus, intersections of both terms indicate peaks and valleys of the probability distribution p[S]. To cover all six possible intersection structures between pcf and pd, we show example snippets for the pd with a variety of curvatures and slopes. (C) The shape of the long-term equilibrium probability distributions (schematically) for the number of synapses of the plastic connection can be derived from the intersection structures in (B): each intersection in (B) leads to a local extremum in the probability distribution in (C). Furthermore there can be peaks at the boundaries. Note, experimental connectivity (Fig. 1A) corresponds to case six which has two intersections. As pcf is monotonically growing, two intersections are only possible for growing pd-functions.