Figure 1.
Synfire chain and its spike activity.
(A) Topology of synfire chain. In a synfire chain, neurons (gray ovals) are organized into successive groups (shown as rows). Each group makes convergent synaptic connections (black arrows) onto the next. Group numbers are shown beside each group. (B) Single trial spike raster of population labeled by group number (upper), and associated population firing rate (lower). Here a group consists of 10 neurons. Each neuron spikes only once, and neurons in a group spike in tight synchrony. The inset shows a detail of the spikes from 3 successive groups. The population firing rate holds steady until the end of the chain, where it drops off to spontaneous levels. (C) Raster plots for select individual neurons across 10 trials. The lowest panel shows a neuron in Group 1, the starting group of the network; it is induced to spike by external input. Successive panels show neurons in higher groups, which spike due to the intrinsic synfire connectivity. The vertical alignment of spikes across trials suggests a high degree of temporal accuracy. The inset shows the details of the spike activity of a neuron in Group 40, which spikes approximately 200 ms after Group 1. (D) A raster plot showing mean spike times (vertical dashs) and spike time jitters (horizontal error bars) for the first 200 neurons across 1000 trials. Insets show the details of groups 3 (lower) and 19 (upper).
Figure 2.
Cartoon of the formation model.
The network is fully connected, but 90% of connections are silent synapses (gray dashed arrows). The active connections (black solid arrows) are randomly set. Black ovals are training neurons (TN), which receive external excitation at the start of each trial, and gray ovals are pool neurons (PN), which spike spontaneously. Following the bent black arrow shows a small subnetwork that includes only the TN and their active synapses and postsynaptic PN. Since the active network is sparse and random, TN do not converge upon the same set of PN except for a random few. Without the ability to turn on silent synapses (follow the gray dashed arrow to the upper middle subnetwork), STDP can act only over the active synapses. Therefore, only the few neurons receiving convergent synaptic input from the TN can spike consistently after the TN. If, however, silent synapses can activate due to spike activity (follow black solid arrow to the lower middle subnetwork), then the TN can activate synapses onto the same set of PN. Since these neurons receive more excitation and hence are more likely to spike, the synapses from the TN to these neurons are more likely to potentiate. This is a positive feedback. These synapses will pass the supersynaptic threshold (follow black solid arrow to the lower right subnetwork), and the TN will coordinate to make convergent synaptic connections onto the same set of PN. The TN do not connect to other neurons due to axon remodeling, in which weak connections from a neuron are pruned once a finite number of super-connections from the same neuron are formed. Without axon remodeling (follow gray dashed arrow to the upper right subnetwork), the TN can continue to activate synapses onto all PN and hoard the entire network to themselves, meaning that all neurons in the network will be induced to spike after they do.
Figure 3.
Topology of supersynapses of a developed network.
Active synapses were originally laid down randomly with a connection probability 0.1. After 200000 trials, the neurons organize into a network that resembles a synfire chain (compare to Figure 1A). Only supersynaptic connections (arrows) and the neurons (circles) that receive them are shown. Light gray circles are saturated neurons; they have withdrawn their axons to all other neurons. Dark gray circles are unsaturated neurons; they have active subsuper synapses that are not shown. Green arrows are synapses that connect to neurons in the next group. Red arrows are synapses that connect to neurons in groups higher than the next. Blue arrows are synapses that connect to neurons in equal or lower groups. Neurons that are labeled in the same group are drawn horizontally in rows; these neurons fire near simultaneously. Successive groups are positioned vertically such that the relative spike time in the network flowing from top to bottom (see Text S1 for details of the algorithm used to assign groups). Each neuron had space to support 10 supersynapses. There were 10 neurons in the training set. Synaptic plasticity parameters for the simulation were set as follows. The LTP constant was GLTP = 0.3; the synaptic conductance threshold for activation/inactivation was ΘA = 0.2; the synaptic conductance threshold for supersynapses was ΘS = 0.4; the maximum synaptic conductance was Gmax = 0.6 (the unit of all conductances is the leak conductance of a neuron). The rate of synaptic decay—the amount by which each synapse is scaled down after every trial—was β = 0.999996. See Materials and Methods for more details.
Figure 4.
Spike timings of neurons in the network shown in Figure 3.
(A) Raster plots (upper) show spike times of neurons for two different stages of the development. Spikes of neurons in the same group are shown in the same row. Spikes of all pool neurons are shown in the same row at the bottom. In trial 50000, there are 21 groups, and the chain activity lasts for approximately 100 ms, after which spontaneous activity of the pool neurons begins. By trial 100000, there are 39 groups, and chain activity lasts about 200 ms. The inset shows a detail of spikes from three successive groups; spikes of a group cluster together, but those of successive groups can overlap. The duration of the chain activity and its growth in time is also demonstrated with the population firing rate (lower). Spikes of all neurons were convolved using a Gaussian kernel with a standard deviation of 3 ms to compute the population firing rate. The firing rates for three different trials—50000, 100000, and 150000 (raster not shown)—are plotted versus time. The duration of chain activity increases linearly with the number of trials (data not shown). (B) Raster plots across trials for select neurons show how precise spike timings emerge. Each panel shows spike data across 300000 trials sampled every 10000th trial for that neuron; the group to which each neuron belongs is indicated in the panel. The neuron in Group 1 is a TN, and therefore is induced to spike at the beginning of each trial (lower panel). The other neurons (upper panels) are recruited into the synfire network at later trials; thus, early on, they only spike spontaneously. As neurons from the chain strengthen synaptic connections onto each neuron, it begins to spike with high accuracy. Inset shows a detail from a neuron in Group 15. (C) Raster plot of mean spike times (vertical dashes) and spike timing jitters (horizontal error bars) for the earliest 100 neurons in the chain. The network—formed over 350000 trials—was simulated for an additional 1000 trials, and the spike data for all neurons was recorded. The first spike time of each neuron was averaged across all 1000 trials, and the jitter (standard error) of the first spike time was calculated. Only those neurons that spiked in at least one-half of all trials are shown. Insets show details from two different time periods. Note that neurons with smaller latency have smaller spike timing jitter.
Figure 5.
The raster shows activity in the network of Figure 5 later in development (Trial 300000). At this point, the network has developed a cycle that replays a portion of the chain activity. Insets show a detail of the same group of neurons in three consecutive cycles.
Figure 6.
Size of developed network versus the total number of neurons.
Simulations were repeated using 500, 1000, 1500, and 2000 neurons. The number of neurons that ended up in the chain were counted and plotted versus the total number of neurons. The relationship is positive.
Figure 7.
Simulations varying the number of training neurons and the number of supersynapses per neuron.
(A) (Upper) Networks formed with three different numbers of training neurons (TN). After a few groups, the width of the synfire network returns to a steady state size. The color coding is identical to Figure 3. (Lower) The distribution of neurons in each group for four networks formed using different numbers of TN; line color and shape encode the different values of number of TN. The number of neurons per group quickly converges to the same number, independent of the number of TN. The inset shows the distribution for the first 7 groups. The curve with 10 TN is from Figure 3; the other three curves are from the networks above. (B) A network formed with the numbers of supersynapses and TN both set to 20; all other parameters were the same as in Figure 3. The major difference compared to the network shown in Figure 3 is that the number of neurons per group is higher.
Figure 8.
The network size as a function of the parameters GLTP and β.
Each square represents a single simulation with the pair of simulation parameters indicated on the axes. For each point, the simulation was run until there were no further changes in the supersynaptic structure. Next, the simulation was run for an additional 100 trials, spike data was collected, and the number of neurons that spike in at least 75 percent of all trials was counted; this number, called the network size, is coded by grayscale and written in each box. A higher value indicates a longer synfire chain network since more neurons are induced to spike regularly.
Figure 9.
Network formation with turnover.
(A) A network formed while neurons died and renewed at an average rate of 1 per 1000 trials through a Poisson process. Even with the turnover, the neurons were able to form a synfire network. (B) Population activity in single trials during the formation process are similar to those without turnover (Figure 4A). The duration of chain activity increases with the number of training trials. (C) Spikes of individual neurons across trials show different behaviors than those without turnover (Figure 4B). A recruited neuron can be deleted (second panel from bottom). Upstream neurons (three upper panels) can shift their spiking times forward as they fill slots vacated by deleted neurons in earlier groups. (D) Spike timings across 100 trials shows that neurons in the chain spike with accuracy on the order of ms.
Figure 10.
Network recovery from a mass lesion.
(A) The mature network of Figure 4 (upper) was given a 20% lesion (middle). The formation process was then allowed to proceed as normal, with no further neuronal death. The network was able recover after 100000 trials (lower); it, however, ended up shorter and wider than normal. (B) Four different simulations using the same base network, but performing different levels of lesions at 10, 20, 30, and 40 percent. The plot shows the change in size of the chain, as defined by the number of neurons spiking reliably, from pre-lesion to post-lesion recovery. The chain does not recover to its normal size; it is shortened. (C) The size of the post-recovery network normalized by the number of neurons left intact directly after the lesion. The normalized size is close to 1 for lesions less than 40%, indicating that neurons are not added nor lost during the recovery. At forty percent however, the normalized size dips below 1, indicating that additional neurons are lost during the recovery period.