N methods, including the separation of time scales not too long ago reviewed in [34, 35], or the use of phenomenological models, for instance FitzHugh agumo (FHN) [36], to regain some degree of mathematical tractability. This has verified especially valuable when studying the response of single neurons to forcing [37], itself a precursor to understanding how networks of interacting neurons can behave. When mediated by synaptic interactions, the repetitive firing of presynaptic neurons may cause oscillatory activation of postsynaptic neurons. In the amount of neural ensembles, synchronised activity of significant numbers of neurons provides rise to macroscopic oscillations, which is often recorded with a micro-electrode embedded within neuronal tissue as a voltage alter known as a neighborhood field prospective (LFP). These oscillations were initial observed outside the brain by Hans Berger in 1924 [38] in electroencephalogram (EEG) recordings, and have offered rise towards the modern classification of brain rhythmsJournal of Mathematical Neuroscience (2016) 6:Web page 5 ofinto frequency bands for alpha activity (eight?3 Hz) (recorded from the occipital lobe through relaxed wakefulness), delta (1? Hz), theta (4? Hz), beta (13?0 Hz) and gamma (30?0 Hz). The latter rhythm is often linked with cognitive processing, and it jp.2015.144 is now common to journal.pone.0073519 hyperlink substantial scale neural oscillations with cognitive states, for instance awareness and consciousness. For instance, from a sensible viewpoint the monitoring of brain states by means of EEG is used to determine depth of anaesthesia [39]. Such macroscopic signals also can arise from interactions in between unique brain places, the thalamo-cortical loop being a classic instance [40]. Neural mass models (describing the coarse grained activity of significant populations of neurons and synapses) have confirmed in particular valuable in understanding EEG rhythms [41], as well as in augmenting the dynamic causal modelling framework (driven by huge scale neuroimaging information) for understanding how event-related responses result in the dynamics of coupled neural populations [42]. A single very influential mathematical approach for analysing networks of neural oscillators, no matter whether they be built from single neuron or neural mass models, has been that of weakly coupled oscillator theory, as comprehensively described by Hoppensteadt and Izhikevich [43]. Inside the limit of weak coupling between limit-cycle oscillators, invariant manifold theory [44] and averaging theory [45] is often applied to decrease the dynamics to a set of phase equations in which the relative phase involving oscillators is the relevant dynamical variable. This strategy has been applied to neural behaviour ranging from that seen in little rhythmic networks [46] as much as the whole brain j.toxlet.2015.11.022 [47]. Regardless of the effective tools and widespread use afforded by this formalism, it does have a quantity of limitations (which include assuming the persistence from the limit cycle below coupling) and it really is effectively to try to remember that there are actually other tools from the mathematical sciences relevant to understanding network behaviour. In this review, we encompass the weakly coupled oscillator formalism inside a range of other approaches ranging from symmetric bifurcation theory as well as the groupoid formalism through to a lot more “O-Propargylpuromycin physics-based” approaches for acquiring reduced models of huge networks. This highlights the regimes exactly where the typical formalism is applicable, and provides a set of complementary tools when it will not.