In this section, we explain why computational models that incorporate learning and maintain stable levels of activity and connectivity have a tendency to enter the overconnected or supercritical state under certain provocations. These provocations include status epilepticus and acute deafferentation, the latter being a model of post-traumatic brain injury. We argue that supercritical connectivity should be epileptogenic.
Recent theoretical and computational work on homeostasis of criticality and activity have done so both in the absence of Hebbian learning [31
], and in its presence [21
]. Perhaps as expected, critical homeostasis is more difficult to achieve in the presence of Hebbian learning, since as we have discussed, learning is a destabilizing force. illustrates our simplest computational model. In this model, a local grouping of neurons is represented by a “node” that can either fire a population spike spontaneously, with no input from any other node, or it can fire in response to activity at one or more other nodes. The latter is referred to as stimulated activity. Most of our simulations were performed with a total of N
= 64 nodes, because our experimental system consists of an 8 by 8 microelectrode array with the four corner electrodes removed (thus, 60 electrodes). At any given time t
, the probability that node i
fires spontaneously within the next time window of 4 ms is given by S
), which can be different for each node and which can also vary in time. At any given time t
, the conditional probability that a prior population spike at node j
causes a population spike at node i
, within the next time window of 4 ms, is given by P
). This conditional probability can be different between every pair of nodes and it can also vary in time. The branching ratio of each node can then be defined as the sum of outputs to all other nodes:
A corresponding measure of excitatory input into a given node i
is given by the input ratio, defined as:
The branching ratio is a pre-synaptic attribute while the input ratio is post-synaptic one; they are equivalent measures of connectivity. Critical connectivity occurs when the branching and input ratios are 1. To achieve critical and firing rate homeostasis, the current firing rate and input ratio for each node i
are compared against the pre-set target firing rate and the critical input ratio of 1. The values of S
) and all the P
)’s associated with that node i
are scaled by small constant factors either up or down so as to approach the target firing rate and critical connectivity. A different factor is used for S
) and the P
)’s. For example, if the current firing rate at node i
is too low compared to the target rate, then S
) and all the P
)’s associated with node i
are scaled up by their respective scaling factors at every time step of the simulation, until the target firing rate is reached or exceeded. If the target firing rate is exceeded, then S
) and all the P
(i, j; t
)’s associated with node i
are scaled down. Similar scaling is also performed for the connectivity, with two different scaling factors, for a total of four different scaling factors [21
Our computational model. Represented are nodes i and j with respective spontaneous firing probabilities S(i) and S(j). The connection strengths are represented by the conditional firing probabilities P(i,j) and P(j,i).
In the presence of Hebbian learning, we found that critical and firing rate homeostasis are independent principles and both
must exist for a neural system to be algorithmically stable. For instance, scaling the S
)’s and P
)’s so as to achieve firing rate homeostasis alone will not guarantee that critical homeostasis is maintained. In addition, there are certain other constraints that must be satisfied for the system to be stable [21
]. One important constraint is that the rate of scaling of the P
)’s must be fast enough to keep up with changes due to Hebbian learning. Another important constraint is that scaling of the P
)’s must operate more quickly than scaling of the S
) ’s. The faster that scaling of the P
)’s takes place, relative to scaling of the S
)’s, the more stable the system. This constraint has important consequences for how networks respond to provocations.
It is important to distinguish spontaneous from stimulated or connectivity-related activity, that is, activity due to the S(i;t)’s vs the P(i,j;t)’s. The importance arises because these two types of activity often do not change in parallel, and in fact, they often change so as to counterbalance each other. Below we give two examples:
Example 1: status epilepticus
shows that forced increased activity of a subset of neurons during a simulated seizure triggers homeostatic mechanisms to scale down all the S(i;t)’s and P(i,j;t)’s to very small values. When the simulated seizure stops, homeostasis causes the S(i;t)’s and P(i,j;t)’s to be scaled back up; they recover to baseline values. However, since scaling of the P(i,j;t)’s must operate more quickly than scaling of the S(i;t)’s, in fact the total connectivity as measured either by the branching or input ratio can overshoot steady state values for a time until the spontaneous firing probabilities, the S(i;t)’s, return to steady state values. Therefore, in the post-ictal state, the overall activity is decreased compared to baseline but the level of connectivity is supercritical. As a result, if and when a population spike occurs, in the post-ictal period, there is an increased chance of abnormally wide spatial spread of this excitation. The significance of activating spatially hyperextended states in a learning system is that if one such hyperextended state occurs frequently enough, the system will “learn” it and “burn” it into memory. If such a state is burned into memory, then there is an increased likelihood that that state will be reactivated again at some random time in the future. The reactivation of a spatially hyperextended state is a necessary condition for epilepsy, as seizures in epilepsy tend to start from the same focus in a stereotypic way, and each seizure focus must involve a macroscopic number of neurons in order to generate clinical symptomatology. Thus we claim that prolonged post-ictal states are epileptogenic, while shorter seizures with no post-ictal state are not as epileptogenic.
Fig 5 Recovery after simulated seizure. A, the spontaneous firing probability declines with seizure onset at arrow, and gradually recovers after seizure stops. B, Connectivity recovers faster than the spontaneous firing probability, and overshoots for 50 million (more ...)
Example 2: acute deafferenation
When a small section of cortex is suddenly deprived of inputs from the rest of the network, epilepsy can gradually develop [49
]. This kind of acute deafferentation is an experimental [53
] and computational model [55
] for post-traumatic epilepsy. In post-traumatic epilepsy in humans, the appearance of epilepsy can be delayed for as long as 10–20 years. What is the mechanism of epileptogenesis and why does it take so long?
We simulated acute deafferentation by allowing 100 nodes to come into equilibrium with each other, and then we suddenly disconnected 10 of these nodes from the others. We then focused attention on the smaller subset of 10 nodes (see ). Our computational model predicts that acute deafferentation causes an immediate drop in firing rate and connectivity in the smaller subset of 10 nodes, because of a loss of excitatory drive from the other 90 nodes. Homeostasis will then cause both connectivity and spontaneous firing probabilities to increase. Connectivity will again adjust more quickly, and indeed it may overshoot. This situation is similar to what happens after prolonged status epilepticus; both status epilepticus and acute deafferentation provoke a hypoactive, hypersynchronous state which can result in the burning into memory of an epileptogenic spatially hyperextended pattern of activation.
Fig 6 Simulated acute deafferentation. A system of 100 nodes is suddenly reduced (arrow) to 10 nodes. A, Steady state spontaneous firing probability is higher for the smaller system. B, Acute deafferentation is accompanied by an immediate drop in the relative (more ...)
In our simulation, the supercritical state is maintained for 9 hours. The time to recovery is longer if the relative magnitude of deafferentation is larger, for instance, by cutting off 20 nodes from a network of one million nodes. In addition, it may be that in real systems, the capacity for increasing the spontaneous activity is limited, such that there is a ceiling above which the spontaneous activity cannot rise. If this ceiling is below what is needed to return connectivity to the critical level, then the deafferentated nodes will remain supercritical indefinitely. Thus it may be that the reason epileptogenesis takes so long is that it is a learned process. It takes time for accidental repetitions of a hyperextended state to burn it into memory.
Our computational result also offers a potential explanation for the results of Graber and Prince [53
], who found that total blockade of all activity by focal application of tetrodotoxin for a period of 3 days, if applied within 3 days after acute deafferentation, can prevent epileptogenesis in rats. Based on our simulations, we suggest that if there is absolutely no activity
for an extended period after acute deafferentation, then no hyperextended states can actually be activated and thus no hyperextended states will be burned into memory, even though connectivity is at supercritical levels. As the spontaneous firing probability gradually approaches or exceeds the new steady state value, if activity is allowed to return to the system at that time, then the connectivity will relax back to critical levels. Once the spontaneous firing probability is near or above its new steady state value, the danger for epileptogenesis has passed.
Therefore, in the prevention of epileptogenesis after acute deafferentation, if one chooses to suppress activity prophylactically, it may be important to suppress activity to a profound degree, because in this time period the connectivity rises to supercritical levels and any activity at all is likely to produce a transient spatially hyperextended state. If one allows these hyperextended states to occur too frequently, by insufficient suppression of activity, then one may actually promote epileptogenesis by teaching the system to burn these hyperextended states into memory. Such a situation would be counterproductive. It would also be important that the method chosen to block activity does not suppress the drive towards higher spontaneous activity. If one suppresses the homeostatic drive to ramp up spontaneous activity, then one may again promote epileptogenesis, in this case by prolonging the epileptogenic period. Such a situation would also be counterproductive.
On the other hand, if the spontaneous firing probabilities can be artificially and more rapidly boosted to near steady state values, then the connectivity should decline more rapidly to its steady state value, because the drive towards higher levels of connectivity has been relieved. That is, boosting spontaneous activity should be protective against epileptogenesis
. This counterintuitive idea arises from analysis of our computational model, and would not have been apparent without such analysis. We have conjectured that this may be one possible mechanism by which electrical brain stimulation works in the treatment of refractory epilepsy. Electrical brain stimulation may work by boosting spontaneous activity and suppressing the supercritical state
To return to the question of homeostatic mechanisms and their relevant timescales, the synaptic scaling mechanism discussed by Turrigiano and colleagues is a slow process, with a timescale of hours to days [44
]. Since critical homeostasis must occur on the same timescale as Hebbian learning, in order to prevent Hebbian learning from destabilizing the system, we must look elsewhere for a fast biomolecular mechanism for critical homeostasis. In addition to being fast, such a mechanism must also be nonlocal, i.e., not restricted to the level of individual synapses, because the input and branching ratios are nonlocal properties requiring simultaneous knowledge of total input and output weights across an entire node. There exists at least three candidate fast, nonlocal mechanisms: (a) When homosynaptic LTP (or LTD) is induced in the intercalated neurons of the amygdala, compensatory heterosynaptic depression (or facilitation) is observed such that the total synaptic weight of a given neuron remains constant [47
]. The counterbalancing heterosynaptic response is suggestive of critical homeostasis. This mechanism depends on the release of intracellular calcium stores and has a timescale of minutes. (b) The phenomenon of backpropagation of action potentials into the dendritic tree [57
] allows widely distributed numbers of synapses to receive nearly simultaneous information about neuronal output. This information is conjectured to play a role in LTP, LTD and STDP, but might conceivably also be used for critical homeostasis. (c) A third mechanism may involve the interaction of principal output neurons with local interneurons. It may be that a certain subset of local interneurons can sense both the total input into and total output out of a local community of output neurons. This information might then be used to modulate either the input or branching ratio of that group of output neurons. In support of this possibility, blocking interneurons with bicuculline can produce a dramatic increase in the branching ratio within minutes (Beggs unpublished).
In summary for this section, the study of the normal function of the brain can provide useful testable hypotheses regarding epileptogenesis. Epileptogenesis is usually discussed in terms of neuronal hyperexcitability, caused by an imbalance between excitatory and inhibitory drives within a neural system or subsystem, although excessive synchronization is also known to play a role [6
]. Within our framework, synchronization is a normal part of brain function, wherein distinct spatial patterns of neuronal activation are activated simultaneously or near-simultaneously to represent an item of stored information. The ability of the brain to maintain a wide repertory of spatial patterns of activation requires an active homeostatic mechanism to maintain neuronal connectivity at the optimal, critical level. Excessive
synchronization can arise when the homeostatic mechanism is provoked in certain ways so as to drive connectivity to supercritical levels. In a learning system, such provocations are epileptogenic because they can lead to the learning and burning into memory of hyperextended spatial patterns of neuronal activation. Thus the role of plasticity in epileptogenesis is not just an unhappy coincidence, a misapplication of the tools of plasticity; rather, the algorithmic requirements of a stable learning system builds in an intrinsic functional vulnerability to epileptogenesis which can be unmasked with repeated provocations.
After neuronal hyperexcitability, the formation of spatially hyperextended states represents the second necessary condition for epileptogenesis. This condition is neither more important nor less important than the condition of neuronal hyperexcitability, but it suggests new ways of thinking about and treating epileptogenic states. For instance, one may wish to maintain connectivity in people at risk for epilepsy more closely about criticality by testing and designing electrical brain stimulation protocols that boost spontaneous activity whenever the brain enters a supercritical state. Suppressing the supercritical state in this way would not only help prevent epileptogenesis but it should also improve brain performance. In contrast, brain stimulation protocols that do not monitor connectivity may not improve brain performance and may not suppress the supercritical state. Similarly, pharmacological suppression of neuronal hyperexcitability also does not guarantee improved brain performance; rather, it more often degrades it.