Interplay between short- and long-term plasticity in cell-assembly formation

Abstract

Various hippocampal and neocortical synapses of mammalian brain show both short-term plasticity and long-term plasticity, which are considered to underlie learning and memory by the brain. According to Hebb's postulate, synaptic plasticity encodes memory traces of past experiences into cell assemblies in cortical circuits. However, it remains unclear how the various forms of long-term and short-term synaptic plasticity cooperatively create and reorganize such cell assemblies. Here, we investigate the mechanism in which the three forms of synaptic plasticity known in cortical circuits, i.e., spike-timing-dependent plasticity (STDP), short-term depression (STD) and homeostatic plasticity, cooperatively generate, retain and reorganize cell assemblies in a recurrent neuronal network model. We show that multiple cell assemblies generated by external stimuli can survive noisy spontaneous network activity for an adequate range of the strength of STD. Furthermore, our model predicts that a symmetric temporal window of STDP, such as observed in dopaminergic modulations on hippocampal neurons, is crucial for the retention and integration of multiple cell assemblies. These results may have implications for the understanding of cortical memory processes.

Figure 1. Rate-dependent plasticity through STDP and homeostatic plasticity.

(A) Spike timing dependence of log-STDP was calculated from equation (7) for given synaptic weights (inset). See Methods for details. (B) Firing rate dependence of synaptic weights at the fixed-point of equation (1) representing synaptic dynamics of STDP and homeostatic plasticity. The fixed weights are analytically calculated for various firing rates of pre-neuron r_pre at given firing rates of post-neuron r_post.

Figure 2. Cell assembly formation by external input for arbitrary strength of STD.

In all panels, “ca” stands for a cell assembly and “bg” for background neurons that do not belong to the assembly. The strength of STD was set as u _sd = 0.1 in simulations from panel B to E. (A) Schematic illustration of the model. We stimulate some of excitatory neurons (blue shaded area) in a randomly connected recurrent neural circuit. Triangles indicate excitatory neurons, whereas circles represent inhibitory neurons. (B) Time evolution of the average synaptic weights within the selected cell assembly (blue), from background excitatory neurons to the assembly (green), from the assembly to background excitatory neurons (cyan), and outside the cell assembly (black). (C) Synaptic weight matrices of excitatory connections are shown before (left) and after (right) the application of external input (arrows in B). Excitatory neurons are separated into 100 bins to calculate the average weights. (D) Raster plots of spiking activity before (left) and after (right) the application of external input, where red dots represent inhibitory spikes and black dots show excitatory spikes. The temporal position of dots are represents the update timing of the spiking state. Neurons 1 to 500 belong to the cell assembly. (E) Dynamics of the average synaptic weight within the cell assembly calculated for various magnitudes of external input I_p. Thin lines are the results from individual simulation trials, and the thick lines are the averages of five simulation trials at each parameter value. (F) Dynamics of the average synaptic weight within the cell assembly calculated at I_p = 1.0 for various values of the release probability u_sd.

Figure 3. Strong STD disturbs cell assembly retention.

(A) Time evolution of average synaptic weights within the selected cell assembly (blue), from background excitatory neurons to the assembly (green), from the assembly to background neurons (cyan), and between background excitatory neurons (black). The left and right panel show results for u_sd = 0.1 and u_sd = 0.5, respectively. (B) Weight matrices of excitatory synaptic connections calculated at t = 30 min are shown for u_sd = 0.1(left) and u_sd = 0.5(right). (C) Raster plots are displayed for the weight matrices shown in B. (D) Dynamics of individual synaptic weights is shown on one excitatory neuron in the assembly. Blue lines correspond to weights from neurons belonging to the assembly, whereas gray lines to those from background excitatory neurons. (E) Distributions of input synaptic weights were calculated from simulation data at t = 26.7–30 min for the neuron shown in D.

Figure 4. Crucial effects of STD on cell assembly retention.

Unless otherwise mentioned, error bars represent the standard deviation obtained by five simulation trials. The results shown in panel A and C to E were calculated at t = 30 min. (A) Relationship between the release probability u_sd and the average synaptic weight within the cell assembly. The results were averaged over five simulation trials. The weights of synapses other than J _EE were constant. (B) Relationship between inhibitory-to-excitatory synaptic weights J_EI and the average firing rates of excitatory neurons is shown in a network model without long-term synaptic plasticity. Horizontal line indicates r_E = 1.8 Hz. (C) Release probability dependence of the average synaptic weight within the assembly is shown. Each plot was calculated using the value of J_EI which sets the average firing rate of excitatory neurons to 1.8 Hz. (D) Relationship between the average synaptic weight within the assembly and input duration is shown. (E) The dependence of the relative synaptic weight w₁ to LTP/LTD ratio g = C_pτ_p/(C_dτ_d), which we varied by changing the value of C_p between 0.015 and 0.0255. (F) Mean-field approximation gives the velocity of weight change as a function of the synaptic weight. Each line is calculated from equation (10) using the steepest descent method from various initial conditions.

Figure 5. Retention of cell assemblies by weak STD.

(A) A first external input activates 20% of excitatory neurons (ca1, blue shaded area), and then a second input successively activates other 20% of excitatory neurons (ca2, green area). Neurons not stimulated by the external inputs are regarded as background (bg). (B) Time evolution of relative synaptic weight w₂. Blue shade indicates the interval of the first stimulus, and the green shade denotes the second one. We defined the retention time of a cell assembly as the time at which w₂ crosses threshold from above (w₂ = 0.015: dotted line). (C) Time evolution of the average synaptic weight for three values of u _sd. The weights were separately averaged over synapses within and between different cell assemblies and background neurons. In the left and middle panels, black lines for bg-to-bg connections are hidden behind purple lines. (D) Raster plots of spiking activity corresponding to the three cases shown in C. Color codes are the same as in Figure 2C . First 500 neurons belong to the first assembly and the second 500 neurons to the second assembly. (E) Synaptic weight matrices of excitatory connections are shown for the above three cases. (F), (G) The relative synaptic weight w₂ and the retention time of ca2 are shown as functions of the release probability u_sd. (H) Relationship between the input duration to ca1 and the relative synaptic weight w₂ at t = 30 min.

Figure 6. STD induces alternate excitations of assemblies, which enlarges synaptic weights within the assemblies.

(A) Null-clines of firing rates for a synaptic weight matrix calculated from equation (9). (B) Potential function U is calculated for the difference in firing rate between two assemblies. The normalization factor U ₀ is determined to ensure U(0) = 0. (C) A monotonic relationship between the release probability and the average interval of the alternation of cell assemblies. The interval was defined as a duration in which one assembly continuously shows higher firing rates than the other. Firing rates were calculated in 10 milliseconds-long time bins. Error bars are the standard derivation of intervals observed during 80 seconds after the stimulus termination in a simulation trial. (D), Typical behavior of the average synaptic weights (above), synaptic efficiency for STD (middle), and neuronal firing rates (below). The first (blue) and second (green) cell assemblies show high firing rates alternately. (E) Relationship between the interval and synaptic weight change for u_sd = 0.1 (cyan) and u_sd = 0.2 (yellow). Inset illustrates the two quantities shown. The ordinate shows synaptic weight change Δw in an interval (Δt_w = 80 milliseconds) starting from the activation of the corresponding cell assembly. Dots are data points obtained from simulation, while solid curves indicate analytic results. (F) Interval dependence of the synaptic weight velocity is shown, which was defined as an expected synaptic weight change in a second. Solid curves show the analytic results calculated at J_ca1 = 0.311, J_ca2 = 0.287, J_bg = 0.156, r_ca1 = 13.38 Hz and r_ca2 = 12.82 Hz.

Figure 7. The retention of cell assemblies with Hebbian and symmetric STDP windows.

(A) An asymmetric STDP window was calculated for J_ij^EE = 0.15. (B) The retention time significantly varies with the release probability of STD. We defined the retention time as a period with a sufficiently large relative weights: w_p>0.1J_EE. (C) Raster plot of spiking activity is shown for the Hebbian STDP rule shown in A. (D) A symmetric STDP window was calculated for J_ij^EE = 0.15. (E) Dynamics of the average synaptic weights at u_sd = 0.2 within (blue) and between (black) assemblies. (F) Raster plot of spiking activity for the symmetric STDP rule shown in D. (G) Relationship between the release probability u_sd and relative weight w_p at t = 30 min. (H) (top) We constructed a histogram of the number of activation over all cell assemblies shown in F. The abscissa shows the number of activation of each assembly normalized by the average number of activation of all assemblies. (middle) We calculated a histogram for the occurrence of all possible 20 (5×4) sequential transitions between two assemblies. The occurrence number of each transition was normalized by the average occurrence number over all transitions. (bottom) Histograms of triplet transitions, such as assembly 1 → 2 → 1 (left) and 1 → 2 → 3 (right), are shown after a normalization by all possible 80 (5×4+5×4×3) triplet transition patterns. All three histograms are obtained from the results of five simulation trials.

Figure 8. Merging and oblivion of cell assemblies through spontaneous activity.

(A) Raster plot of spiking activity in a network embedding 32 cell assemblies. Active epochs of initial assemblies are shown by different colors in the left panel, while those of merged assemblies are shown in the right panel. (B) Synaptic weight matrix after 30 minutes of spontaneous activity. (C) A graphical representation of the merged connection matrix, where each numbered circle corresponds to an initial assembly. (D) Relationship between the storage capacity and the release probability. (E) The survival rate of each assembly depends on the initial magnitudes of intra-assembly synaptic weights. We separated cell assemblies into four groups according to the initial weight values ( : the boundaries were decided such that each group contains 5 to 15 assemblies) and calculated the fraction of the assemblies that survived in the reorganization. See Methods for other details of the simulations. (F) The rate of merging of a cell assembly as a function of the initial synaptic weight. As in E, we separated 992 inter-assembly connections into five groups ( ) so that each group contains more than 100 assemblies.