Behavioral Time Scale Plasticity of Place Fields: Mathematical Analysis

Abstract

Traditional synaptic plasticity experiments and models depend on tight temporal correlations between pre- and postsynaptic activity. These tight temporal correlations, on the order of tens of milliseconds, are incompatible with significantly longer behavioral time scales, and as such might not be able to account for plasticity induced by behavior. Indeed, recent findings in hippocampus suggest that rapid, bidirectional synaptic plasticity which modifies place fields in CA1 operates at behavioral time scales. These experimental results suggest that presynaptic activity generates synaptic eligibility traces both for potentiation and depression, which last on the order of seconds. These traces can be converted to changes in synaptic efficacies by the activation of an instructive signal that depends on naturally occurring or experimentally induced plateau potentials. We have developed a simple mathematical model that is consistent with these observations. This model can be fully analyzed to find the fixed points of induced place fields and how these fixed points depend on system parameters such as the size and shape of presynaptic place fields, the animal's velocity during induction, and the parameters of the plasticity rule. We also make predictions about the convergence time to these fixed points, both for induced and pre-existing place fields.

Keywords: eligibility trace; hippocampus; mathematical model; place field; synaptic plasticity.

Figure 1

Running track and model network. (A) A mouse runs at velocity v along a running track with locations marked by unique features. Inside the mouse hippocampus, N CA3 place cells have activity peaks at different locations along the track, and synapse onto a single postsynaptic CA1 cell. (B) The CA3 place cells considered here are modeled as simple Gaussians centered at evenly spaced locations along the running track.

Figure 2

Weight dynamics. (A) An illustration of the synaptic plasticity traces and the instructive signal. (B) A simple dynamical model of synaptic plasticity, where both LTP and LTD require an overlap between a trace variable T^k and an instructive signal P. W is the active synaptic weight, and W_in are internalized resources, the total synaptic weight is conserved. (C) The overlap I^k between the traces and the instructive signal, as a function of D, where D = t_P − t₀ is the displacement between the start of the instructive signal and the center of the presynaptic place field in units of time.

Figure 3

Fixed point place field structure. (A–C) Fixed point weights as a function of D for different sets of parameters. The velocity in all these subplots is identical (v = 0.116 m/s, chosen to match the velocity in Milstein et al., 2020), and the same presynaptic Gaussian place field is used, with a SD width of 0.21 m. All other parameters are indicated above each subplot.

Figure 4

Place field plasticity on a linear track. (A,B) Simulated evolution of ramp amplitude over laps (dashed lines) and the numerically calculated fixed point ramp amplitude (solid lines) as a function of D. For (A), initial weights are set to zero. For (B), weights are initialized such that there is a preexisting place field at D = 7.0 s. (C) Change in ramp amplitude as a function of D and the initial ramp amplitude, for 10 different simulations. To replicate experiment, each simulation has a random selection of initial place field strength, plateau potential strength, and plateau potential location. These response curves replicate the results of Figure 1J of Milstein et al. (2020), with the caveat that our response curves are smoother and more uniform since we are assuming a uniform velocity.

Figure 5

Place field plasticity on a circular track. (A) Traces and instructive signal for a lap >> 1 on the circular track, as a function of time. The presynaptic traces plotted are from a CA3 input centered at location of the instructive signal. (B) I_p and I_d on circular track. (C) Fixed point for weights on the circular track, compared to fixed point weights for the linear track. Notice also that (A) is plotted in absolute time, where as (B,C) are plotted in terms of D (time relative to instructive signal onset). The solution in the circular track is nearly symmetric around D = 0, while the linear solution contains goes to zero for D < −0.5s due to its lack of periodic boundary conditions.

Figure 6

Place fields with non-linear traces. (A) In an example non-linear case, the effective presynaptic activity profiles that drive LTP and LTD traces are different. Here the LTD traces are driven by the linear RF as above (black) but LTP traces are driven by a non-linear modification of the RF (dashed-red). (B) LTP (blue) and LTD (orange) traces in the non-linear case. The instructive signal in dashed green. (C) The resulting fixed point of the weight vector. Results here are for V = 0.15 m/s.

Figure 7

The dependence of place field shape on velocity during induction. (A) Fixed points of the ramp amplitude as a function of D_x for differing movement velocities, circular track. Inset: velocities during induction, in m/s. (B) Same as (A), but for a linear track. The lack of periodic boundary conditions introduces some asymmetry in the fixed point ramp amplitudes. (C) The dependence of the half width of the weights at fixed-point on the travel velocity during induction along a circular track. The green and orange symbols are for left and right widths, respectively.

Figure 8

Convergence to solutions. (A) The relative distance to the fixed point ($\frac{W_{fixed}-W}{W_{fixed}}$) as a function of D and trial number, shown as logarithmic heatmap. (B) The convergence time, τ_w (blue), and the number of simulated trials to reach $\frac{W_{fixed}}{e}$ away from the fixed point (orange) as a function of D (in seconds). Notably, in both cases the convergence time rises steeply beyond a certain value of D. Both cases start from the initial condition W_i = 0.

Figure 9

Analytical solutions for a rectangular receptive field. (A) I_p, I_d, and (B) W_fixed as a function of D. Here, the LTD trace has a basal level $T_0^d$, which creates an additional overlap $\gamma T_0^d{\tau }_I(1-e^{\frac{t_P-t_{t}rial}{{\tau }_I}})$ (see Appendix). The resulting fixed point is nearly symmetrical around D = 0, and its properties can be modified via the model parameters. The track is linear, and the parameters used for the figure are as follows: τ_p = 500 ms, τ_d = 1,500 ms, η_p = 0.25, η_d = 200, $T_{max}^p$ = 2.2, $T_{max}^d$ = 2.0, $T_0^p$ = 0, $T_0^d$ = 1.5.