Christoph Kirst

Dynamic information routing in complex networks

Flexible information routing fundamentally underlies the function of many biological and artificial networks. Yet, how such systems may specifically communicate and dynamically route information is not well understood. Here we identify a generic mechanism to route information on top of collective dynamical reference states in complex networks. Switching between collective dynamics induces flexible reorganization of information sharing and routing patterns, as quantified by delayed mutual information and transfer entropy measures between activities of a networks units. We demonstrate the power of this mechanism specifically for oscillatory dynamics and analyse how individual unit properties, the network topology and external inputs co-act to systematically organize information routing. For multi-scale, modular architectures, we resolve routing patterns at all levels. Interestingly, local interventions within one sub-network may remotely determine nonlocal network-wide communication. These results help understanding and designing information routing patterns across systems where collective dynamics co-occurs with a communication function.

Guaranteeing global synchronization in networks with stochastic interactions

We design the interactions between oscillators communicating via variably delayed pulse coupling to guarantee their synchronization on arbitrary network topologies. We identify a class of response functions and prove convergence to network-wide synchrony from arbitrary initial conditions. Synchrony is achieved if the pulse emission is unreliable or intentionally probabilistic. These results support the design of scalable, reliable and energy- efficient communication protocols for fully distributed synchronization as needed, e.g., in mobile phone networks, embedded systems, sensor networks and autonomously interacting swarm robots.

Sequential Desynchronization in Networks of Spiking Neurons with Partial Reset

The response of a neuron to synaptic input strongly depends on whether or not the neuron has just emitted a spike. We propose a neuron model that after spike emission exhibits a partial response to residual input charges and study its collective network dynamics analytically. We uncover a desynchronization mechanism that causes a sequential desynchronization transition: In globally coupled neurons an increase in the strength of the partial response induces a sequence of bifurcations from states with large clusters of synchronously firing neurons, through states with smaller clusters to completely asynchronous spiking. We briefly discuss key consequences of this mechanism for more general networks of biophysical neurons.

How precise is the timing of action potentials

Distributed spiking activity underlies the dynamics and function of neuronal circuits and thus their computational capabilities. Beyond a simple rate, often the timing of spikes also essentially contributes to information processing in these systems (Hahnloser et al., 2002; Riehle et al., 1996; Rieke et al., 1996; Rokem et al., 2006). A thorough understanding and analysis of the very notion of spike timing is therefore pivotal for understanding brain function. For instance, it is widely accepted that abstract discrete time models of interacting neurons, with spike times fixed to a temporal grid, may well describe the spike rates of neurons, e.g., for balanced cortical activity (van Vreeswijk and Sompolinsky, 1996), but the timing of spikes is not modeled exactly. In this respect, even simple integrate-and-fire type models are more accurate because they describe neural dynamics in continuous time and thus may exhibit spikes at any chosen time (Brette et al., 2007). One may argue that the manually implemented reset in integrate-and-fire models, leading to exact spike times, merely serves as a low-level compromise between detailed biological modeling and mathematical tractability. Generating an action potential takes a time of the order of 1 ms, but one may easily introduce a defined time of a spike by interpreting it, e.g., as the time of peak voltage during a biophysical action potential (cf. Figure ​Figure11). Figure 1 Definition of spike timing (A) in models with threshold and reset and (B) in models with active spike generation. (A) Membrane potential V of a leaky integrate-and-fire model receiving excitatory input current I; when V crosses the threshold (dashed line) ... Nevertheless, continuous time models, in general, face the conceptual problem that information contained in the timing of only a single spike is infinite. In contrast, discrete time models exhibit bounds on the information carried by a spike but it may seem questionable how real biological systems would conform to time discretization. Moreover, in raster plots displaying experimentally recorded spike trains of neurons there is actually a raster, a non-zero time resolution discretizing time into small but positive intervals. However, the current high temporal resolution, often 10 kHz or more, may make us forget such discretization issues. In their recent contribution to Frontiers in Neuroscience, Cessac and Vieville (2008) emphasize that the main issue is not about how fine the resolution actually is, in models or data, but whether or not there is a discretization at all. So not even the most subtle description, neither experimentally nor in modeling, can characterize the timing of spikes with arbitrarily high precision. The authors now show an alternative way of modeling spiking neural circuits by lifting a recent mathematical work (Cessac, 2008) to the level of networks with conductance-based synapses and by pointing out (and explicitly highlighting for their system) a number of prerequisites needed to fully grasp what neural network modeling is all about – how to define “spikes” and what is their “timing”, how could we come by conceptual problems of discrete resolution and how well do (arbitrarily detailed) mathematical models characterize an actual neural systems’ dynamics. They combine continuous time evolution with a discretization of spike times and identify situations where minute disturbances in spike times may crucially change the circuit dynamics. That work certainly does not provide the final word on the subject, but highlights one key problem: it is not fully self-evident a priori how precise the timing of action potentials really is and what we actually mean by “timing”, neither in experimental data nor in idealized mathematical models. Cessac and Vieville (2008) argue that the very notion of spike timing is not well understood in itself and thus can lead to conceptual difficulties. For instance, in integrate-and-fire type models the reset implies that the neurons membrane potential after reset is completely independent of its value before reset. A very recent work (Kirst et al., 2009) on a state- and input-dependent reset partially resolves this issue. One might expect that the problem of spike timing is overcome completely when considering biophysically more detailed models, such as Hodgkin–Huxley or compartment models; but even for arbitrarily refined, high-dimensional differential equation models, any reasonable time scale described must be much larger than intrinsic time scales characterizing, e.g., individual ion channels, because otherwise the very description by differential equations looses its meaning. The study of Cessac and Vieville (2008), pushing further an alternative discrete-time view onto the world of biological neural network modeling, naturally raises more questions than it answers: in their model, discrete spike times themselves are defined arbitrarily precise (namely on the lattice) such that it remains debatable in how far the above precision problem is actually solved. More generally, how does noise affect the spike timing in networks and what is the impact of the dynamics of action potential initiation (cf. Naundorf et al., 2006)? We also need to reconsider related questions about creating (or removing) additional spikes by small perturbations and about the reliability of spikes (Jahnke et al., 2008; Teramae and Fukai, 2008). For computations in neural systems it finally seems most relevant how precisely spike times can actually be detected by neurons and read out for further processing (Tiesinga et al., 2008). We definitely need to take some time to precisely think about timing before recording, simulating or analyzing the timing of action potentials in neural circuits.

From networks of unstable attractors to heteroclinic switching.

We present a dynamical system that naturally exhibits two unstable attractors that are completely enclosed by each others basin volume. This counterintuitive phenomenon occurs in networks of pulse-coupled oscillators with delayed interactions. We analytically show that upon continuously removing a local noninvertibility of the system, the two unstable attractors become a set of two nonattracting saddle states that are heteroclinically connected. This transition equally occurs from larger networks of unstable attractors to heteroclinic structures and constitutes a new type of singular bifurcation in dynamical systems.

Neuromodulatory Control of Long-Term Behavioral Patterns and Individuality across Development

Animals generate complex patterns of behavior across development that may be shared or unique to individuals. Here, we examine the contributions of developmental programs and individual variation to behavior by monitoring single Caenorhabditis elegans nematodes over their complete developmental trajectories and quantifying their behavior at high spatiotemporal resolution. These measurements reveal reproducible trajectories of spontaneous foraging behaviors that are stereotyped within and between developmental stages. Dopamine, serotonin, the neuropeptide receptor NPR-1, and the TGF-β peptide DAF-7 each have stage-specific effects on behavioral trajectories, implying the existence of a modular temporal program controlled by neuromodulators. In addition, a fraction of individuals within isogenic populations raised in controlled environments have consistent, non-genetic behavioral biases that persist across development. Several neuromodulatory systems increase or decrease the degree of non-genetic individuality to shape sustained patterns of behavior across the population.

Convergence of Self-Organizing Pulse-Coupled Oscillator Synchronization in Dynamic Networks

The theory of pulse-coupled oscillators provides a framework to formulate and develop self-organizing synchronization strategies for wireless communications and mobile computing. These strategies show low complexity and are adaptive to changes in the network. Even though several protocols have been proposed and theoretical insight was gained there is no proof that guarantees synchronization of the oscillator phases in general dynamic coupling topologies under technological constraints. Here, we introduce a family of coupling strategies for pulse-coupled oscillators and prove that synchronization emerges for systems with arbitrary connected and dynamic topologies, individually changing signal propagation and processing delays, and stochastic pulse emission. It is shown by simulations how unreliable links or intentionally incomplete communication between oscillators can improve synchronization performance.

Partial Reset in Pulse-Coupled Oscillators

Pulse-coupled threshold units serve as paradigmatic models for a wide range of complex systems. When the state variable of a unit crosses a threshold, the unit sends a pulse that is received by other units, thereby mediating the interactions. At the same time, the state variable of the sending unit is reset. Here we present and analyze a class of pulse-coupled oscillators where the reset may be partial only and is mediated by a partial reset function. Such a partial reset characterizes intrinsic physical or biophysical features of a unit, e.g., resistive coupling between dendrite and soma of compartmental neurons; at the same time the description in terms of a partial reset enables a rigorous mathematical investigation of the collective network dynamics. The partial reset acts as a desynchronization mechanism. For N all-to-all pulse-coupled oscillators an increase in the strength of the partial reset causes a sequence of desynchronizing bifurcations from the fully synchronous state via states with large c...

Local control of non-local information flow in oscillatory neuronal networks

Control of information flow between neurons or groups of neurons is essential in a functional brain, e.g. for context and brain state dependent processing. In line with recent experimental and theoretical studies [1-5] we show that phase relations between synchronized oscillatory local circuits or brain areas may dynamically create information channels and induce changes in the effective connectivity. Reducing neuronal oscillatory dynamics to a phase - amplitude description [6,7], we show how alternative phase shifts between different neurons or groups of neurons result in different effective connectivities. In particular, to quantify the information flow, we analytically calculate the time delayed mutual information and transfer entropy between oscillators in a phase locked state. We further present a theoretical framework to predict phase lag patterns within and between groups of oscillators in hierarchical networks. Combining both results we derive the information flow between the oscillators as a function of structural and dynamical network parameter. We use our results to reveal how effective connectivity is controlled by the underlying physical connectivity and the intrinsic single oscillation frequencies. Interestingly, we find that local changes in the strength of a single link can remotely control the effective connectivity between two different physically unchanged oscillators. Similarly, local inputs modulating the intrinsic frequencies can dynamically and remotely change the information flow between distal nodes. We link our results to biophysically more realistic networks of spiking neurons. In a clustered network of groups of type I neurons exhibiting gamma oscillations emanating from a PING mechanism [8], we numerically show that local changes of the connectivity or the inputs strengths within a cluster can non-locally control the phase relations and the information flow between distant clusters.

hPSC Modeling Reveals that Fate Selection of Cortical Deep Projection Neurons Occurs in the Subplate

Cortical deep projection neurons (DPNs) are implicated in neurodevelopmental disorders. Although recent findings emphasize post-mitotic programs in projection neuron fate selection, the establishment of primate DPN identity during layer formation is not well understood. The subplate lies underneath the developing cortex and is a post-mitotic compartment that is transiently and disproportionately enlarged in primates in the second trimester. The evolutionary significance of subplate expansion, the molecular identity of its neurons, and its contribution to primate corticogenesis remain open questions. By modeling subplate formation with human pluripotent stem cells (hPSCs), we show that all classes of cortical DPNs can be specified from subplate neurons (SPNs). Post-mitotic WNT signaling regulates DPN class selection, and DPNs in the caudal fetal cortex appear to exclusively derive from SPNs. Our findings indicate that SPNs have evolved in primates as an important source of DPNs that contribute to cortical lamination prior to their known role in circuit formation.