Shortest Loops are Pacemakers in Random Networks of Electrically Coupled Axons

Abstract

High-frequency oscillations (HFOs) are an important part of brain activity in health and disease. However, their origins remain obscure and controversial. One possible mechanism depends on the presence of sparsely distributed gap junctions that electrically couple the axons of principal cells. A plexus of electrically coupled axons is modeled as a random network with bi-directional connections between its nodes. Under certain conditions the network can demonstrate one of two types of oscillatory activity. Type I oscillations (100–200 Hz) are predicted to be caused by spontaneously spiking axons in a network with strong (high conductance) gap junctions. Type II oscillations (200–300 Hz) require no spontaneous spiking and relatively weak (low-conductance) gap junctions, across which spike propagation failures occur. The type II oscillations are reentrant and self-sustained. Here we examine what determines the frequency of type II oscillations. Using simulations we show that the distribution of loop lengths is the key factor for determining frequency in type II network oscillations. We first analyze spike failure between two electrically coupled cells using a model of anatomically reconstructed CA1 pyramidal neuron. Then network oscillations are studied by a cellular automaton model with random network connectivity, in which we control loop statistics. We show that oscillation periods can be predicted from the network’s loop statistics. The shortest loop, around which a spike can travel, is the most likely pacemaker candidate. The principle of one loop as a pacemaker is remarkable, because random networks contain a large number of loops juxtaposed and superimposed, and their number rapidly grows with network size. This principle allows us to predict the frequency of oscillations from network connectivity and visa versa. We finally propose that type I oscillations may correspond to ripples, while type II oscillations correspond to so-called fast ripples.

Gap junction networks can generate both ripple‐like and fast ripple‐like oscillations

“…A conceptual understanding of how re‐entry can work in a complicated network that contains multiple cycles comes from the work of Vladimirov et al . () and Singh et al . () (see also Gansert et al ., ).…”

“…actually succeeds in establishing re‐entry. We see that one step of propagation is present (into cells 116 and 133), but do not show the precise complete path that leads to a successful return to cell 98 – a matter of some complexity (see Vladimirov et al ., ). Computational analysis of the present network structure showed that there are 2698 distinct cycles that contain both hub cell 68 and hub cell 98; thus, there are many possibilities available.…”

“…When cyclic (‘circus’) activity is modeled in abstract, cellular‐automaton networks, either with neuronal‐like (Traub et al ., , ; Lewis & Rinzel, ; Vladimirov et al ., , ) or with more general abstract elements (Singh et al ., ), it becomes clear that the minimal cycle length is equal to the total number of ‘refractory’ plus ‘resting’ plus ‘active’ states. ‘Refractoriness’ in a cellular‐automaton model, however, does not have a precise correlate in more physiologically realistic neuronal models with Hodgkin–Huxley‐like ionic conductances.…”

In situ three-dimensional reconstruction of mouse heart sympathetic innervation by two-photon excitation fluorescence imaging

“…Thus, the presence of multiple varicosities that are distributed in an ‘ en passant ’, bead-on-a-string manner along thin axonal branches as well as axonal branching are expected to significantly impact on axial conduction and, thus, to modify signal propagation within the network. Also, the formation of nested-loop axonal trajectories in the infarct border zone may promote high-frequency electrical reentry within the local network, giving rise to abnormal sympathetic signaling (Vladimirov et al, 2012). …”

Waves and Patterns on Regular Graphs

“…In [23], the authors use a pyramidal cell model and also an excitable cellular automata model to study the possibility of high-frequency oscillations in randomly connected graphs. Their random graphs employ scalefree networks where the degree of each node is slightly less than 2.…”