# Theory of Precise Timing in Spiking Neural Networks

Patterns of precisely timed and spatially distributed spikes have been experimentally observed in different neuronal systems [1]. These spike patterns correlate with external stimuli (events) and are thus considered key features of neural computation [2]. Their dynamical origin, however, is unclear. One possible explanation for their occurrence is the existence of excitatorily coupled feed-forward structures, synfire chains [3] which are embedded in a network of otherwise random connectivity and receive a large number of random external inputs. In complementary theoretical modeling studies we investigate whether precise spike timing and temporal locking can naturally arise in the nonlinear dynamics of recurrent neural networks that contain no additionally embedded feed-forward structures. In particular, we study how strongly heterogeneous networks of complicated connectivity can still coordinate the timing of spikes of different neurons, possibly even if they are not directly linked by a synapse. We are addressing different groups of phenomena:

**Speed Limits to Coordinating Spike Times - Revealed by Random Matrix Theory **(in collaboration with F. Wolf and T. Geisel, Dept. Nonlinear Dynamics). In large networks of spiking neurons, we uncovered a topology-induced speed limit to coordinating spike times [4] and explained both the speed and its limit via random matrix theory. The mechanism underlying the coordinating barrier points to a cooperative effect of the existence of a minimum time a neuron needs to generate subsequent spikes, the temporally discrete communication between neurons and their complex interaction network [5].

**Network Design: Heterogeneous Networks with Specific Dynamics. **Under which conditions can spiking neural networks exhibit certain predefined patterns of precisely timed spikes? For a broad class of model networks (that include, among other features, strong heterogeneities, complicated connectivity and distributed delays) we analytically determined the subset of all networks (parameterized by features of single neurons and their interactions) which exhibit an arbitrary predefined pattern [6]. This provides a novel method to find potential network structures, for instance by modifying the synaptic interaction strengths that generate a desired, e.g. experimentally observed dynamics. In particular, the method enables us to find stable as well as unstable patterns, which might both be computationally relevant, cf. [7]. In a first application, we combined this network design with additional requirements on the resulting network, for instance realizing networks with a given type of coupling (e.g., only inhibitory) and a given type of connectivity (e.g., with exponentially distributed number synapses per neuron) which simultaneously exhibits a desired spike pattern. In a second application, we identified networks that exhibit a predefined pattern, but simultaneously optimize structural features, e.g. minimize wiring costs [8]. Recent experimental results [9] obtained by the group of W. Singer, Max Planck Institute for Brain Research, Frankfurt, strongly indicate the importance of the precise timing of spikes in unprecedented detail. This initiated a further intense investigation of the origin of precise timing in recurrent networks. Furthermore, in collaboration with the BCCN Berlin and the RIKEN Brain Science Institute, we recently developed a novel method for detecting precise timing dependencies between spiking activity and the coarser signal of local field potentials [M. Denker et al., in prep.]. Application of the method to activity recorded from motor cortex of awake behaving monkeys revealed that during a movement preparation task spikes tend to keep a fixed phase relationship to the LFP, largely independent of the LFP amplitude [10].

**How Chaotic is the Balanced State? **Highly irregular dynamics is a prominent feature of multi-dimensional complex systems and often attributed to chaos. In particular, sparsely coupled networks of spiking neurons exhibit a balanced state [11] with very irregular dynamics. Here excitatory and inhibitory inputs to each neuron balance on average and only the fluctuations create spikes at irregular, seemingly random points in time. Mean field theory shows that such balanced activity occurs in networks with excitatory and inhibitory recurrent feedback as well as in networks that receive external excitatory inputs and exhibit recurrent inhibition only. Going beyond mean field theory, our studies on the microscopic dynamics of the latter networks reveals that the dynamics in fact is not chaotic but rather dynamically stable [12], a fact that comes as a surprise to many theoreticians, not only in the neurosciences.

[1] M. Abeles et al. (1993) *J Neurophysiol *70(4):1629-1638; Y. Ikegaya et al. (2004) *Science* 304:559-564

[2] M. Abeles (2004) *Science* 304:523-524

[3] M. Diesmann, M.-O. Gewaltig, and A. Aertsen (1999) *Nature* 402:529-533; Y. Aviel, C. Mehring, M. Abeles, and D. Horn (2003) *Neural Comput* 15:1321-1340

[4] M. Timme, F. Wolf, and T. Geisel (2004) *Phys Rev Lett* 92:074101

[5] M. Timme, T. Geisel, and F. Wolf (2006) *Chaos* 16:015108

[6] R.-M. Memmesheimer, M. Timme, and T. Geisel (2005) In: *Verhandl. DPG (VI)* 40, 2/2005, DY 22.6, pg 202; R.-M. Memmesheimer and M. Timme (2006) *Phys Rev Lett *97:188101

[7] P. Ashwin and M. Timme (2005) *Nonlinearity* 18:2035-2060 and (2005) *Nature* 436:36-37

[8] R.-M. Memmesheimer and M. Timme (2006) *Physica D* 224:182-201

[9] K. Gansel and W. Singer (2005) *Soc Neurosci Abstr* 276.8

[10] M. Denker, S. Roux, M. Timme, A. Riehle, S. Grün (2007) *Neurocomputing* 70:2096

[11] C. van Vreeswijk and H. Sompolinsky (1996) * Science* 274:1724

[12] S. Jahnke R.-M. Memmesheimer, and M. Timme *Phys Rev Lett *100:048102 (2008).

**Contact:**Marc Timme

### Members working within this Project:

Marc TimmeChristian Tetzlaff

Diemut Regel

#### Former Members:

Christoph KirstRaoul Martin Memmesheimer

Hinrich Arnoldt

Fabio Schittler Neves

Birgit Kriener

Andreas Sorge

Annette Witt