Eigenvalues of block structured asymmetric random matrices

We study the spectrum of an asymmetric random matrix with block structured variances. The rows and columns of the random square matrix are divided into $D$ partitions with arbitrary size (linear in $N$). The parameters of the model are the variances of elements in each block, summarized in $g\in\mathbb{R}^{D\times D}_+$. Using the Hermitization approach and by studying the matrix-valued Stieltjes transform we show that these matrices have a circularly symmetric spectrum, we give an explicit formula for their spectral radius and a set of implicit equations for the full density function. We discuss applications of this model to neural networks

Chaos in heterogeneous neural networks: I. The critical transition point

There is accumulating evidence that biological neural networks possess optimal computational capacity when they are at or near a critical point in which the network transitions to a chaotic regime. We derive a formula for the critical point of a general heterogeneous neural network. This formula relates the structure of the network to its critical point. The heterogeneity of the network may describe the spatial structure, a multiplicity of cell types or any selective connectivity rules

Chaos in heterogeneous neural networks: II. Multiple activity modes

We study the activity of a recurrent neural network consisting of multiple cell groups through the structure of its correlations by showing how the rules that govern the strengths of connections between the different cell groups shape the average autocorrelation found in each group. We derive an analytical expression for the number of independent autocorrelation modes the network can concurrently sustain. Each mode corresponds to a non-zero component of the network’s autocorrelation, when it is projected on a specific set of basis vectors. In a companion abstract we derive a formula for the first mode, and hence the entire network, to become active. When the network is just above the critical point where it becomes active all groups of cells have the same autocorrelation function up to a constant multiplicative factor. We derive here a formula for this multiplicative factor which is in fact the ratio of the average firing rate of each group. As the effective synaptic gain grows a second activity mode appears, the autocorrelation functions of each group have different shapes, and the network becomes doubly chaotic. We generalize this result to understand how many modes of activity can be found in a heterogeneous network based on its connectivity structure. Finally, we use our theory to understand the dynamics of a clustered network where cells from the same group are strongly connected compared to cells from different groups. We show how this structure can lead to a one or more activity modes and interesting switching effects in the identity of the dominant cluster

Light That Appears to Come from a Source That Does Not Exist

Superoscillatory, band-limited functions oscillate faster than their fastest Fourier component. Superoscillations have been intensively explored recently as they give rise to many out-of-the-spectrum phenomena entailing both fundamental and applied significance. We experimentally demonstrate a form of superoscillations which is manifested by light apparently coming from a source located far away from the actual one. These superoscillations are sensed through sharp transverse shifts in the local wave vector at the minima of a pinhole diffraction pattern. We call this phenomenon “optical ventriloquism.

25th annual computational neuroscience meeting: CNS-2016

The same neuron may play different functional roles in the neural circuits to which it belongs. For example, neurons in the Tritonia pedal ganglia may participate in variable phases of the swim motor rhythms [1]. While such neuronal functional variability is likely to play a major role the delivery of the functionality of neural systems, it is difficult to study it in most nervous systems. We work on the pyloric rhythm network of the crustacean stomatogastric ganglion (STG) [2]. Typically network models of the STG treat neurons of the same functional type as a single model neuron (e.g. PD neurons), assuming the same conductance parameters for these neurons and implying their synchronous firing [3, 4]. However, simultaneous recording of PD neurons shows differences between the timings of spikes of these neurons. This may indicate functional variability of these neurons. Here we modelled separately the two PD neurons of the STG in a multi-neuron model of the pyloric network. Our neuron models comply with known correlations between conductance parameters of ionic currents. Our results reproduce the experimental finding of increasing spike time distance between spikes originating from the two model PD neurons during their synchronised burst phase. The PD neuron with the larger calcium conductance generates its spikes before the other PD neuron. Larger potassium conductance values in the follower neuron imply longer delays between spikes, see Fig. 17.Neuromodulators change the conductance parameters of neurons and maintain the ratios of these parameters [5]. Our results show that such changes may shift the individual contribution of two PD neurons to the PD-phase of the pyloric rhythm altering their functionality within this rhythm. Our work paves the way towards an accessible experimental and computational framework for the analysis of the mechanisms and impact of functional variability of neurons within the neural circuits to which they belong