Figure 1.
Experimentally obtained distributions of the number of synapses on a single connection and the here proposed structural plasticity model.
(A) The relative frequencies p[S] of the number of synapses between two neurons obtained from experiments [10, 36–39] show strong peaks for no synapses and 3–8 synapses. The probability for no synapses (S = 0) reaches from 0.75 to 0.99 (for [38] this probability was taken from [81]). (B) Scheme of the proposed structural plasticity model shows relevant quantities of the model for one connection. Synapses are created from a pool of potential synapses Pij with a constant probability pbuild. Each of the Sij realised synapses has a weight wij,k (k = 1 … Sij) which develops according to a synaptic plasticity rule. Synapse removal happens with a weight-dependent probability pdel(wij,k). All influences which do not relate to the examined neurons are modelled by currents Ii or Ij. Inset: An example for the weight-dependency of building and removal probabilities shows that smaller synapses are more likely to be deleted. (See main text for more details)
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.
Table 1.
Distribution shapes for different numbers of extrema.
Figure 3.
Most of the commonly used rate-based learning rules do not provide a positive correlation between weight and postsynaptic activity.
The slope of the pd-term in Equation 2 is determined by the slope of the fixed weight depending on postsynaptic activity ( relation) resulting from the synaptic plasticity rule. Here we show the relation of commonly used learning rules [18] for the simple feedforward system (top row) and for a linear approximation of a feedback system, where the presynaptic activity equals the postsynaptic activity (bottom row). For reproducing experimental data, pd has to grow, i.e. the fixed weights have to increase with postsynaptic activity. This is fulfilled (red shaded area), for example, by the BCM-learning rule with synaptic scaling or a Hebb-like learning rule with synaptic scaling and feedback but also for more biological rules like the calcium-based plasticity rule from [24] (see Materials and Methods for parameters used to generate this figure).
Figure 4.
Model can account for experimental data for suitable pre- and postsynaptic activities.
(A) The probability distribution of the number of synapses between two neurons from experiment ([36], red) is similar to the distribution resulting from the proposed model (blue) at vj = 0.656 and vi(S = 0) = 0.2975. (B) The activity confidence regions, where error of the experimental outcome lies within the most probable 95% (yellow) or 66% (green) of the trials, when randomly sampling from model distribution, spans over a broad range of activities. (C) Colour code shows the Monte-Carlo p-values for the hypothesis that experimental data comes from model distribution. For comparability, postsynaptic influence I was transformed to the postsynaptic activity for S = 0 in all figures. (Parameters: BCM rule with synaptic scaling with θ = 0.08, vtss = 0.1, κ = 9.0, structural plasticity: P = 12, ln pbuild = −16, a = 2.0, ρ = 0.125)
Figure 5.
Different experimental data can be explained at different activity regions; effects are robust to underlying distributions or parameter changes.
(A) The 95% confidence regions for experimental data from different cortical layers ([10, 36, 37], Fig. 1A) are located at different activities for different layers. Confidence regions for same experimental location (layer II) overlay. (B) The 95% confidence regions, which emerge from the same model parameters when using a distribution of potential synapses from [40], are qualitatively not different. (C), Schematic drawing which summarises the influence of a and ρ on the location of the confidence region (see (E) and (F)). For the layer IV data, we evaluated how (D) the area in activity space, (E) the average presynaptic activity, and (F) the average postsynaptic influence of the activity confidence region changes for different structural plasticity parameters a and ρ. (Parameters BCM rule with synaptic scaling with θ = 0.08, vtss = 0.1, κ = 9.0, structural plasticity: P = 12, ln pbuild = −16, in A-C: a = 2.0, ρ = 0.125)
Figure 6.
Quantitative estimates of the activities show biological reasonable ordering in a reasonable frequency regime.
(A) Activity confidence regions for intra-layer connections in somatosensory cortex. To obtain quantitative results, we used an input-output-relation (blue line) from an adaptive exponential integrate-and-fire neuron [54] and a calcium-based plasticity rule [24] (see Methods). As in Fig. 5, the activities for layer IV connections are larger than those for layer II connections. The firing frequencies lie in a biological reasonable range. (B) The expectation value of the postsynaptic activity for stimulations within the confidence intervals confirm that experimental data can be reproduced with equal pre- and postsynaptic activity as expected for intra-layer connections.
Figure 7.
The BCM rule feedforward connection shows a hysteresis in pre- and postsynaptic stimulation.
(A) The predicted probability distributions p[S] (in colour-code) for the system from Fig. 4A is strongly influenced by varying the postsynaptic stimulation. Black lines indicate the values of S (treated as continuous variable) for which synapse creation and deletion are equally probable. These points correspond to stable (continuous line) or unstable (dashed line) fixed points of the dynamical system following the net probability flow of our system and indicate the existence of local extrema in the long-term equilibrium probability distribution. Two bifurcations lead to an appearance and disappearance of a bistability, which indicates a possible hysteresis. (B) The same applies for varying the presynaptic activity, although the second bifurcation on the right hand side does not reveal for continuous S, but takes place in the discrete case as both sign changes happen between two consecutive states. (C) Simulation reveals the predicted hysteresis: postsynaptic stimulation was increased stepwise such that vi(S = 0) increased by steps of 0.01 until it reached 1.0 and then decreased again. For each stimulation, the average number of synapses was calculated separately for the in- and decreasing direction and later averaged over all stimulation cycles (see Methods). The blue curve depicts the average number of synapses in the increasing and the green curve the decreasing direction. (D) Altering presynaptic activity in the same way also yields a hysteresis loop. (Parameters: BCM rule with synaptic scaling with μ = 0.2, θ = 0.08; υtss = 0.1; κ = 9.0, structural plasticity P = 12; ln pbuild = −16, a = 2.0, ρ = 0.125, in A, C: vj = 0.656, in B, D: vi(S = 0) = 0.2975)
Table 2.
Rate-based synaptic plasticity rules and their -dependencies.