Article Figures & Data
Figures
- Figure 1.
Small-world network. a, Networks of neurons are generated in which all cells are only coupled to their nearest neighbors (4 in this case). b, To generate small-world networks, small numbers of connections are broken and rewired to make long-distance connections at random locations. Long-distance connections reduce the number of synapses between any pair of neurons in the network. c, As more long-distance connections are added, the network loses the property that most connections are local, and the network looks much more random. We find a range of normal and epileptiform behaviors in the small-world network regimen, where few connections are necessary to connect any pair of the neurons, but local connections still predominate.
- Figure 3.
Transition from normal → seizing → bursting behavior as a function of the number of long-distance connections (ρ). The left column shows the results from the CA1 model with N = 3000 and k = 30, whereas the right column shows the results from a CA3 model with N = 3000 and k = 90. At the top are three examples of data (taken from a Poisson-simulated network) for normal, seizing, and bursting, showing the count of neurons that fired in a 10 msec time bin. Middle panels illustrate the total population activity for Poisson (Poiss), noisy leaky integrate-and-fire (IF), and stochastic Hodgkin-Huxley (HH) simulations with the examples from above indicated by a-f. Vertical bars indicate boundaries between normal, seizing, and bursting as identified by eye from the time traces of population activity from the Poisson model. Simulations of IF network with 24,000 neurons and reduced synaptic strength are displayed as well (IF-24k). These networks show qualitative behaviors similar to the 3000 neurons. The bottom panels illustrate the normalized clustering coefficients and mean path length between neurons in the network as the proportion of long-distance connections in the network (ρ) is increased. The left side of these graphs indicates a network topology in which the ring of neurons has only local connections; the right side indicates a nearly randomly connected network.
- Figure 2.
Bursting and seizing behaviors as the number of long-distance connections are changed. a, The ring contains N neurons, each of which are connected to k, mostly local neighbors (left). To visualize the activity of this large network, we color coded each point according to the state of the neuron and pulled every kth point in the ring toward the center to make a spoke. Therefore, a neuron in the center of a spoke is connected to all the neurons in the spoke, assuming that all synaptic connections are local. A neuron at the end of the spoke is connected to half of the neurons on the spoke and half of the neurons on the opposite end of the next spoke. This results in a plot of the ring that resembles a slinky. b, An illustrative temporal snapshot of network activity, with N = 3000, k = 30 synapses per neuron (i.e., 1% network connectivity), and ρ = 0.1. Light gray dots represent excitable neurons, black dots are firing neurons, and dark gray dots are refractory neurons. The wave front size stabilizes to approximately half the size of the local neighborhood k and is followed by a refractory tail. This tail is determined by how many steps the wave front can travel before the neurons begin to recover. c, Successive frames from a movie of seizing activity, with N = 3000, k = 30, and ρ = 0.1 (i.e., that 10% of synapses have been rewired). The frame rate is 250 Hz, corresponding to approximately two synaptic time delays; therefore, the active waves appear twice as large (in space) as their actual size. Spontaneous background activity generates a cascade of activity, which stabilizes into two traveling waves (frames 5-25). These traveling waves generate other waves in the network through the long-distance connections (e.g., frames 26, 31, 34). Eventually, waves start to meet and annihilate each other (e.g., frames 4, 33, 43). This network attains equilibrium when the new waves are generated at the same rate that the waves annihilate each other. d, Still frames from a movie of bursting activity (N = 3000; k = 90; ρ = 0.1). In this network, the number of long-distance connections causes waves to generate new waves faster than the waves annihilate each other. This results in all of the neurons firing in the network, all of the neurons becoming refractory, and the activity in the network shutting off. Movies of network activity can be seen at: http://www.bu.edu/ndl/people/netoff/SWN/JNeurosciSupplement.html.
- Figure 4.
The birth- death process (1-dimensional map) model of wave generation and annihilation. The top panels show how many new waves are generated (ni; dashed lines) and annihilated (di; solid lines) per time step as functions of the number of currently active waves, with k = 90. The y-intercept of the dashed lines indicates the spontaneous background rate of wave generation. An equilibrium point exists where the new wave rate is equal to the dying wave rate (indicated by the arrows). The bottom panels map the number of waves on one time step to the average number expected on the next time step (solid lines). Equilibria occur when the number of waves on the next time step is equal to that on the current time step [i.e., at the intersection of the solid line of f(wi) and the dotted line of identity (also indicated by arrows)]. Forρ = 0.0001, the equilibrium point at wi ≈ 1.82 is only weakly attracting (the slope of the solid line is approximately + 1), and the number of waves changes only slightly at each time step. Forρ = 0.01, the system has a strongly attracting equilibrium (where the slope of the solid line is approximately zero), corresponding to ongoing seizures characterized by an average of 3.87 existing simultaneous waves. For ρ = 0.05, the equilibrium is unstable, and the dynamics has entered a chaotic regimen. The onset of chaos indicates that the entire network will repeatedly fire brief synchronous bursts.
- Figure 5.
Change in network behavior as a function of number of synaptic connections per neuron and proportion of long-distance connections using the reduced model (for network size of 3000 neurons). The left panel shows curves delineating the normal, seizing, and bursting regimens as the number of synapses per neuron, and the proportion of long-distance connections are changed. The solid black curve is calculated from analysis of the one-dimensional map and the dotted black curve from the (1 + R)-dimensional map. Tick marks indicate the boundaries of normal, seizing, and bursting behavior in network simulations from Figure 3. Horizontal lines indicate specific parameter choices for the CA3 and the CA1 models. Points labeled a-f correspond to the conditions simulated in Figure 3a-f. These plots imply that the CA3 network will transition from normal to bursting at a much smaller proportion of long-distance connections or smaller synaptic strength than the CA1. The right panel illustrates the boundaries between the behavioral regimens as the number of synapses per neuron and the synaptic strength are varied (with the proportion of long-distance connections fixed at ρ = 0.01). The curves were calculated in a similar way to that in the left panel. The tick mark in the line for the CA3 indicates the boundary between bursting and seizing observed in the integrate-and-fire model, where the population firing rate as a function of synaptic strength are plotted in the inset. The results of the simulations correlate well with the analyses of the reduced map models. No clear transition from seizing to bursting was seen in the full CA1 model, as predicted by the one-dimensional map model.
Tables
Symbol Identification Definition/values used N Number of neurons in network 3000 (24,000 in some network simulations) k Number of synapses per neuron 30 (for CA1), 90 (for CA3) s Spontaneous firing rate of a single neuron per time step of size τd 0.0315 × τd τR Absolute refractory time of neuron 28 msec (IF), 36 msec (Poiss, HH) τd Synaptic time delay 2.8 msec (IF), 3.7 msec (Poiss), 1-5 msec (HH) ρ Proportion of long-distance connections generated by breaking a synapse and rewiring it to a randomly chosen postsynaptic cell Varied from 1.0 × 10−5 to 0.4 p1 Synaptic strength (i.e., probability that postsynaptic neuron will fire given that a particular presynaptic neuron fired) 0.025 p2 Probability that two postsynaptic neurons fire given the presynaptic neuron fired (dependent on k and p1) p2 = 1 - (1 - p1)k - kp1(1 - p1)k−1 α Approximate number of neurons in wave front k/2 - 1 R Number of time steps that a neuron remains refractory R = τR/τd ≈ 10 Wi Number of waves present in network at time i Wi + 1 = f(Wi) ei Number of excitable neurons in the network at time i 
or ei = N − αwi(1+R) (for one-dimensional map) ni Number of new waves generated at time i resulting from long-distance connections ni = (2αWi kρ)(p1p2ei/N) + Si di Number of waves that die in time step i resulting from wave collision di = 2αWi/ei Si Spontaneous wave generation resulting from spontaneous cellular activity Si = seip2 W* Number of waves in network where new wave rate and dying wave rate are equal (equilibrium point) f(W*) = W* f(Wi) Function describing number of waves on next time step given number of waves on time step i f(Wi) = Wi + ni − di
