The Journal of Neuroscience, December 3, 2003, 23(35):11167-11177
Materials and Methods
In the present study, we identified a new mode of spontaneous activity in cortical networks from organotypic cultures andacute slices: the neuronal avalanche. Neuronal avalanches were characterized by three distinct findings: (1) Propagation of synchronized LFP activity was described by a power law. (2) The slope of this power law, as well as the branching parameter, indicate that the mechanism underlying theseavalanches is a critical branching process. (3) Our network simulations and pharmacological experiments suggest that a critical branching process optimizes information transmission while preserving stability in cortical networks.
The analysis presented here focuses exclusively on the propagation of sharp (<20> commonly observed in slice cultures (Jimbo and Robinson, 2000) or evoked extracellular potentials in acute slices. Current source density analysis in combination with optical recordings has demonstrated that sharp, negative LFPs peaks are indicative of synchronized population spikes (Plenz and Aertsen, 1993).Similarly, cortical LFPs in vivo are closely correlated with single spike cross-correlations of local neuronal populations (Arieli, 1992). Thus, negative LFP peaks in the present study might represent synchronized action potentials from local neuronal populations. This is supported by computer simulations of the neuron-electrode junction of planar microelectrode arrays, which demonstrate that sharp, negative LFPs originate from synchronizedaction potentials from neurons within the vicinity of the electrode (Bove et al., 1996). Therefore, our results might be specifically applicable to the propagation of synchronized action potentials in the form of neuronal avalanches through the network, and the power law of -3/2 provides the statistical framework for transmitting information through the cortical network in form of locally synchronized action potential volleys.
Other authors who have studied propagation of synchronized action potentials in neural networks have concluded that precise patterns of activity could travel through several synaptic stages without much attenuation (Abeles, 1992; Aertsen et al., 1996; Reyes, 2003). The concept of a critical branching process does not necessarily conflict with this view, but does place constraints on the distance that activity could propagate when it is traveling in avalanche form. Although it is natural to think that a critical branching parameter of 1 will produce a sequence of neural activity in which one neuron activates only one other neuron at every time step, this is not the case. Because the branching parameter reflects a statistical average, it gives only the expected number of descendants after many branching events, not the exact number at every event. Thus, a single neuron might activate more than one other neuron on some occasions, whereas on others it may activate none. In fact, the most common outcome in the critical state will be that no other neurons are activated. The resulting events generated by this system will contain many short avalanches, some medium-sized avalanches, and very few large avalanches.
The neuronal activity discussed here has numerous points of contact with this body of theory: (1) All cortical networks displayed power law distributions of avalanche sizes. (2) The cortical networks in the cultures arrived at this state without any external instructive signal. (3) The slope of the power law for avalanche sizes and for avalanche life times, as well as the experimentally obtained values of all indicate that the avalanches can be accurately modeled as a critical branching process. For these reasons, the activity observed in the cortical networks should be considered as neuronalavalanches.
The power law in the present study basically says that the number of avalanches observed in the data scales with the size of the avalanche, raised to the -1.5 power. This allows for a prediction of very large avalanches. They are a natural consequence of the local rule for optimized propagation, and are expected to occur even in normal (i.e., nonepileptic) networks, and are not particularly rare. For example, in a network with 10,000avalanches/hr that engage just one electrode, at least 21 avalanches will occur every hour that will encompass exactly all 60 electrodes. Thus, on average, activity on every electrode will be correlated with every other electrode in the network at least once every 3 min.
The neuronal avalanches described here are profoundly different from previously observed modes of network operation. As shown by the correlograms, activity in the cortical networks was not periodic or oscillatory within the duration of maximal avalanche lifetimes. In addition, the contiguity index revealed that activity at one electrode most often skipped over the nearest neighbors, indicating that propagation was not wave-like. Finally, although the spontaneous activity did display notable synchrony at relatively long time scales, the avalanches that we describe here actuallyoccurred within such synchronous epochs at a much shorter time scale (<100>avalanches themselves did not display synchrony, regardless of the threshold level, IED, or number of electrodes used to obtain the data. These are compelling reasons for neuronalavalanches to be considered a new mode of network activity.
Previous theoretical work has discussed the importance of a balance between excitation and inhibition in network dynamics (Van Vreeswijk and Sompolinsky, 1996; Shadlen and Newsome, 1998). This balance has been implicated in proportional amplification in cortical networks (Douglas et al., 1995) as well as in the maintenance of cortical up states (Shu et al., 2003). Here, we extend the idea of balance by using the branching parameter, a concept that allows us to explore information transmission at the network level. Although a branching parameter well below unity would confer stability on a network, the simulations suggest that this stability would come at the rather severe price of greatly reduced information transmission. In contrast, a branching parameter hovering near unity would optimize information transmission, but at the risk of losing stability every time the network became supercritical. Although these neural network simulations are vastly oversimplified representations of the dynamics that occur in cortical networksin vivo, they may nonetheless offer some insight as to why the cerebral cortex is so often at risk for developing epilepsy. In fact, our experimental results demonstrate that removal of inhibition to increase propagation in the neuronal network to obtain a power law with slope > -1.5 results in epileptic activity. The competing demands of stability and information transmission may both be satisfied in a network whose branching parameter is at or slightly below the critical value of 1. Thus, calculating the power law exponent and/or branching parameter might offer quantitative means to evaluatethe efficacy of cortical networks to transmit information.