Branching processes are widely used to model phenomena from networks to neuronal avalanching. In a large class of continuous-time branching processes, we study the temporal scaling of the moments of the instant population size, the survival probability, expected avalanche duration, the so-called avalanche shape, the n-point correlation function and the probability density function of the total avalanche size. Previous studies have shown universality in certain observables of branching processes using probabilistic arguments, however, a comprehensive description is lacking. We derive the field theory that describes the process and demonstrate how to use it to calculate the relevant observables and their scaling to leading order in time, revealing the universality of the moments of the population size. Our results explain why the first and second moment of the offspring distribution are sufficient to fully characterise the process in the vicinity of criticality, regardless of the underlying offspring distribution. This finding implies that branching processes are universal. We illustrate our analytical results with computer simulations. *

Carrier scattering is known to crucially affect the dynamics of quantum dot (QD) laser devices. We show that the dynamic properties of a QD laser under optical injection are also affected by Coulomb scattering processes and can be optimized by band structure engineering. The nonlinear dynamics of optically injected QD lasers is numerically analyzed as a function of microscopically calculated scattering lifetimes. These lifetimes alter the turn-on damping of the solitary QD laser as well as the complex bifurcation scenarios of the laser under optical injection. Furthermore, we find a pump current sensitivity of the frequency-locking range, which is directly related to the nonlinearity of the carrier lifetimes.

Recently, neuronal avalanches have been observed to display oscillations, a phenomenon regarded as the coexistence of a scale-free behaviour (the avalanches close to criticality) and scale-dependent dynamics (the oscillations). ordinary continuous-time branching processes with constant extinction and branching rates are commonly used as models of neuronal activity, yet they lack any such timedependence. In the present work, we extend a basic branching process by allowing the extinction rate to oscillate in time as a new model to describe cortical dynamics. By means of a perturbative field theory, we derive relevant observables in closed form. We support our findings by quantitative comparison to numerics and qualitative comparison to available experimental results. In the brain, electrical signals propagate between neurons of the cortical network through action potentials, which are spikes of polarisation in the membrane of the neuron's axon. These spikes have an amplitude of about 100 mV, typically last about 1 ms 1 and can be recorded using micro-electrodes 2,3. In order to study the signaling in larger regions of neurons, multielectrode arrays, comprising about 60 electrodes spread across ≈ 4 mm 2 , are used to capture the collective occurrence of spikes as local field potentials (LFPs). In this setting, the electrodes are extracellular and each is sensitive to electrical signals from several surrounding neurons 4-8. The data of the LFP recordings are processed by putting them into time bins of a few milliseconds duration and by introducing a voltage threshold. In addition, a refractory period is imposed to avoid counting large voltage excursions more than once. The details of processing can differ slightly between experiments 5-9. However, after processing, the data is a time series of two values for each electrode: on (detected signal above threshold) and off (no detected signal or signal below threshold). A neuronal avalanche is then defined as a set of uninterrupted signals detected across the micro-electrode array. Each avalanche is both preceded and succeeded by at least one time bin where none of the electrodes detected a signal, defining the avalanche duration as the number of time bins where the avalanche unfolds. Which and how many electrodes detect a signal varies during the avalanche 5,6. The duration of avalanches typically ranges between a few milliseconds and 30 ms 10. A prominent observable is the avalanche size, which is the total number of recorded signals during the avalanche. If there is only one electrode detecting a signal in each time bin of an avalanche, its size and duration are equal. However, the size of an avalanche is usually larger than its duration due to the simultaneous detection of signals by different electrodes. Histograms of the avalanche size show an apparent power-law distribution of sizes, with common small avalanches and rare large avalanches 5,6 , the fingerprint of scale-free phenomena. The exponent of the power law was observed to be −3/2 in Ref. 5. However, its exac...

Inside cells of living organisms, actin filaments and microtubules self-assemble and dissemble dynamically by incorporating actin or tubulin from the cell plasma or releasing it into their tips' surroundings. Such reaction-diusion systems can show diusion-or reaction-limited behaviour. However, neither limit explains the experimental data: while the oset of the linear relation between growth speed and bulk tubulin density contradicts the diusion limit, the surprisingly large variance of the growth speed rejects a pure reaction limit. In this article, we accommodate both limits and use a Doi-Peliti field-theory model to estimate how diusive transport is perturbing the chemical reactions at the filament tip. Furthermore, a crossover bulk density is predicted at which the limiting process changes from chemical reactions to diusive transport. In addition, we explain and estimate larger variances of the growth speed.