Synaptic plasticity in neural networks needs homeostasis with a fast rate detector

Abstract

Hebbian changes of excitatory synapses are driven by and further enhance correlations between pre- and postsynaptic activities. Hence, Hebbian plasticity forms a positive feedback loop that can lead to instability in simulated neural networks. To keep activity at healthy, low levels, plasticity must therefore incorporate homeostatic control mechanisms. We find in numerical simulations of recurrent networks with a realistic triplet-based spike-timing-dependent plasticity rule (triplet STDP) that homeostasis has to detect rate changes on a timescale of seconds to minutes to keep the activity stable. We confirm this result in a generic mean-field formulation of network activity and homeostatic plasticity. Our results strongly suggest the existence of a homeostatic regulatory mechanism that reacts to firing rate changes on the order of seconds to minutes.

Figure 1. The balanced network model.

(A) Schematic of the network model. Recurrent synapses in the population of excitatory neurons (*) are subject to the homeostatic triplet STDP rule. (B) Typical magnitude and time course of a single excitatory postsynaptic potential from rest. (C) Membrane potential trace of a cell during background activity. (D) Histogram of single neuron firing rates (blue) and coefficient of variation (CV ISI, red) across neurons as well as the ISI distribution of all neurons (yellow) of the network during background activity. Arrowheads indicate mean values.

Figure 2. Network stability during ongoing synaptic plasticity depends crucially on the homeostatic time constant.

(A) Temporal evolution of the average firing rate in the excitatory population for different homeostatic time constants . Explosion of firing rate indicated by dashed lines. Curves for (dark blue), (light blue), and (turquoise) overlap on the interval from 2 h to 24 h indicating stability. With (black) we show one of the cases with very short where the activity spontaneously dies. (B) Spike raster of 200 randomly selected excitatory neurons. The last two seconds are shown before the network activity destabilizes (). (C) For , the activity stays asynchronous and irregular even after 24 h hours of simulated time. (D) Firing statistics in a stable network () measured after 24 h of simulated time. Histogram of single neuron firing rates (blue) and coefficient of variation (CV ISI, red) across neurons and the ISI distribution of all neurons (yellow). Arrowheads indicate mean values. Black lines represent the corresponding statistics prior to any synaptic modifications (copied from Figure 1). (E) Population firing rate for stable simulation runs at as a function of the homeostatic time constant. The dashed line indicates the target firing rate . (F) Evolution of the synaptic weight distribution during the first 8 hours of synaptic plasticity ().

Figure 3. Mean field theory predicts the stability of background activity.

(A) Schematic of the mean field model. Plastic synapses are indicated by *. (B) Eigenvalues of the Jacobian evaluated at the non-trivial fixed point . (C) Phase portrait for , a choice where background activity is stable. Nullclines are drawn in black. Arrows indicate the direction of the flow. Two prototypical trajectories starting close to are shown. Blue line: Typical example of a solution that returns to the stable fixed point. Solutions starting in the shaded area, such as the red line, diverge to infinity. (D) The separatrix for four different values of . (E) Population firing rate of the spiking network model (simulations: red dots) for different values of weight for connections from excitatory to excitatory neurons. Black line: Least-square fit of Eq. (3) on the interval as indicated by the black bar. Extracted parameters are and (cf. Eq. (3)).

Figure 4. The mean field predictions agree with results from direct simulation of the spiking network.

(A) Solid line: as a function of the learning rate (cf. Eq. (7)), with simulation data (red points) for . The arrow indicates the value used throughout the rest of this figure (the dotted line corresponds to the learning rate as used in Figure S1). (B) Same as before but as a function of for fixed. (C) Lifetime values for the spiking network (red points) with a scaled step function as predicted by mean field theory ( and ). All error bars are smaller than the data points.

Figure 5. Slow synaptic weight decay renders weight distribution unimodal, but hardly affects global stability.

(A) Evolution of the synaptic weight distribution over 8 h of background activity. (B) Synaptic weight distribution at . (C) Predictions for of mean field theory (solid line) and values obtained from direct simulation (points). (D) Final population firing rate as a function of for values of where the background state is a stable fixed point (dashed line: target rate ; error bars: standard deviation over 100 bins of 1 s).

Figure 6. Triplet STDP with synaptic scaling requires a fast rate detector.

(A) Black line: Eigenvalues of the Jacobian () for different values of (). Gray curve: Values from Figure 3 B for reference. The red line (“sim”) indicates the critical value as obtained from simulating the full spiking network. (B) As before, but for different values of (). (C) Lifetimes of the background state in simulated networks of spiking neurons for different values of (). (D) Phase plane with nullclines. -nullcline in black; -nullclines: dashed (), gray () and red (). The latter was used in the rest of the figure. (E) Synaptic weight distribution after of simulation.

Figure 7. Postsynaptic priming affects STDP protocols.

Simulation of the metaplastic triplet STDP rule . (A) Top: Typical protocol for the induction of LTD (75 pairs (post-pre) at 5 Hz with −10 ms spike offset) in the triplet STDP model () with a postsynaptic cell which is quiescent prior to the LTD protocol (black) compared to induction after postsynaptic priming (blue). Top, left: Pre- and postsynaptic spikes for priming and. Top, right: LTD induction. Middle: postsynaptic rate estimate of the postsynaptic cell. Bottom: Weight change over time. Postsynaptic priming period (duration 100 s): regular firing at terminated by one second of silence () to avoid triplet effects. (B) Relative differences in final weight change between quiet () and primed protocol () at the end of a LTD (gray) plasticity protocol. LTP protocol for reference (hollow, same paring protocol, with reversed timing, +10 ms spike offset). Left: For different durations of the priming period and fixed priming frequency of 3 Hz. Right: Different priming frequencies with fixed priming duration of 60 s. The black line is a RMS fit to LTD data points of: (left) an exponential function; (right) of a quadratic function.