A nonlinear cable framework for bidirectional synaptic plasticity

Abstract

Finding the rules underlying how axons of cortical neurons form neural circuits and modify their corresponding synaptic strength is the still subject of intense research. Experiments have shown that internal calcium concentration, and both the precise timing and temporal order of pre and postsynaptic action potentials, are important constituents governing whether the strength of a synapse located on the dendrite is increased or decreased. In particular, previous investigations focusing on spike timing-dependent plasticity (STDP) have typically observed an asymmetric temporal window governing changes in synaptic efficacy. Such a temporal window emphasizes that if a presynaptic spike, arriving at the synaptic terminal, precedes the generation of a postsynaptic action potential, then the synapse is potentiated; however if the temporal order is reversed, then depression occurs. Furthermore, recent experimental studies have now demonstrated that the temporal window also depends on the dendritic location of the synapse. Specifically, it was shown that in distal regions of the apical dendrite, the magnitude of potentiation was smaller and the window for depression was broader, when compared to observations from the proximal region of the dendrite. To date, the underlying mechanism(s) for such a distance-dependent effect is (are) currently unknown. Here, using the ionic cable theory framework in conjunction with the standard calcium based plasticity model, we show for the first time that such distance-dependent inhomogeneities in the temporal learning window for STDP can be largely explained by both the spatial and active properties of the dendrite.

Figure 1

(A): The calcium dependent function implements the calcium control hypothesis, when no change in the synaptic weight occurs, for synaptic depression (LTD) occurs, and for synaptic weights are increased (LTP) (parameters used were , , , , . B: Calcium dependent learning rate (adapted from [45]). Parameters were , , , .

Figure 2. Resulting STDP learning windows taken from two different positions along the cable.

In (A) and (C), note how the illustrated plasticity outcomes for a single pre- and postsynaptic pairing using Eqn. (2), derived from the calcium control hypothesis using calcium based plasticity rule, changes as a function of distance. In particular, comparing (A) to (C) note the increase in the time constant of the LTD portion of the STDP window when the position from the source of spike initiation is increased. This behavior mimics the location-dependent nature of the STDP window as observed in (C) and (D) as presented in previous physiological experiment . Parameters used were , , , , , , , , and . (Figs 2B and 2D were adapted from [31]).

Figure 3. Simple changes to morphology by changing the diameter results in visible changes to the STDP window taken from two different positions along the cable.

(A) For the first location at , it seems that the influence of varying cable diameter is negligible but for (B) the second position at , we observe more prominent changes to the STDP window, in particular the disappearance of LTD portion of the window for decreasing diameters. This is mainly caused through larger normalized calcium concentration values that are used to calculate synaptic weight changes.

Figure 4. Changes in excitability through changes in ion channel hotspot density can change the location dependence of the STDP window to different degrees.

(A) shows how increasing the density of sodium channels by had little change on the profile of the STDP window for the first location at , but for (B) the second location at there is an evident change in the STDP window where the LTD contribution has all but vanished for the increased sodium channel density. This is attributed to the increased calcium influx into the location of the dendrite. (C) and (D) illustrate that varying the density of high-voltage activated had little impact on both the profiles and the spatial dependence of the STDP window. (E) and (F) shows that increasing the density of low-voltage activated calcium channels by gave rise to similar changes in the profile (and spatial dependence) of the STDP window at the two pre-specified locations, however for smaller variations few differences were observed.

Figure 5. The outcomes of pairing frequency effects taken from Sjöström's original experiment and the simulation.

In A, we present the original data by Sjöström for comparison with our simulation results. In B, we display the outcome of Shouval's calcium based plasticity rule where changes are driven by normalized peak calcium. Note how the resulting plasticity profiles for and as a function of frequency lack a notable crossover point at 40 Hz. In C and D, again CaDP is used but this is driven by a modified calcium-dependent variable where the ratio between the peak calcium and the maximal integral over time of the calcium concentration profile is used to drive plastic change. In C and D, one can observe distance dependent changes in the plasticity outcomes for the pairing frequency protocol (as explained in text). In C, we see that the msec curve gives rise to only LTP, while for the msec there is an initial region that gives rise to LTD but then at a particular frequency there is a switch from LTD to LTP. Furthermore, note that the two curves crossover at 42 Hz. This is similar to Sjöström's experimental findings in A. In D, one can see a rightward shift of the msec curve where it inherits a small LTD component, however for the msec data there is a notable rightward shift of the LTD region leading to increase in frequency when LTD switches to LTP.

Figure 6. A schematic illustration of a dendritic cable studded with ion channel hotspots.

A single hotspot is represented as a black band and denotes an infinitesimal area on the cable, containing ion channels of type where single ion channels are represented as red spots.