# Attractor basins of stable state sequences in balanced circuits of spiking neurons

In the brain, information is processed by dynamically generated sequences of circuit activity states encompassing thousands of neurons. Although fundamental to information processing, the phase space structure of circuits of spiking neurons remains poorly understood. Recently, a collective phase of cortex-like circuit activity was discovered [1] that naturally partitions the phase space into a rich set of diverging attractor basins of complex stable network state trajectories. We have developed a theory for the geometry and scaling properties of their attractor basins. We find they are structured by the preimages of future spike crossing events involving particular network connectivity motifs. We examine the statistical mechanics of their random geometry by introducing and averaging a function dependent on the microstate with zeros at the basin boundaries. This theory provides a foundation for the rational design of spiking circuits that use intrinsic stable spiking patterns for neuronal computation.

Figure: (A) Neuron index versus time showing spike times from of all neurons of the given attracting trajectory, *φ*(*t*), in a 150 ms window (*N = 200, K = 50*). (B) 2+1D folded phase space volume, (*δφ*_{1}*, δφ*_{2}*, t*)*,* centered around *φ*(*t*) located at (0, 0, *t*) (black line). The center basin is filled grey, and the two cross-sections, (*δφ*_{1}*, δφ*_{2}*, *0) and (*δφ*_{1}*, δφ*_{2}*, *150) are shown. The basin size decays exponentially but can undergo abrupt jumps at spikes, e.g. *s*_{i} and *s*_{i+1}, corresponding to crossings of spikes from neurons that share a connection. (C) The fraction of restored perturbations, *f _{R}*, as a function of the perturbation strength, ε, for different values of the coupling strength,

*J*

_{0}. (D) A schematic illustration of the boundary contracting with the phase space until the crossing event and the boundary flagging function,

*f*.

_{T}**Contact:**Fred Wolf

### Members working within this Project:

Fred Wolf#### Former Members:

Rainer EngelkenMaximilian Puelma Touzel