Encyclopedia of Computational Neuroscience

Living Edition
| Editors: Dieter Jaeger, Ranu Jung

Slow Oscillations: Models

  • Albert CompteEmail author
Living reference work entry
DOI: https://doi.org/10.1007/978-1-4614-7320-6_307-1


Firing Rate Cortical Network Slow Oscillation Persistent Sodium Current Thalamocortical Neuron 
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Several computational models have been proposed to describe the generation of spontaneous slow oscillations, or up and down state dynamics, in neuronal circuits of the cerebral cortex. Most models rely on strong excitatory feedback to generate reverberatory dynamics in the up state, which are then quenched by some negative feedback mechanism to form down states. The specific mechanisms responsible for triggering the transitions between up and down states distinguish these models. In addition, some models have been proposed that emphasize the role of thalamocortical loops in generating this spontaneous activity.

Detailed Description

During non-REM sleep and under most anesthetics, cortical network dynamics exhibit slow oscillations or up and down state switching (see Slow-Oscillations: Physiology). This pattern of activity has been the subject of numerous computational models, which have formulated quantitative accounts of the possible mechanisms that participate in their generation.

The models contemplate two main mechanisms to ignite up states. One set of models proposes that down states end as a result of the recovery of a fatigue mechanism accumulated during up states, such as some activity-dependent potassium current (Compte et al. 2003; Destexhe 2009), or short-term synaptic depression (Mejias et al. 2010). Spontaneously active excitatory neurons, possibly in specific layers (Destexhe 2009), are then responsible for starting a new up state. This links down state duration to the time constant of recovery of such fatigue mechanism. On the other hand, up states could be started by input fluctuations that impose a transition from a stable down state to the up state. Mechanistically, these fluctuations could be due to spontaneous synaptic release (Timofeev et al. 2000; Holcman and Tsodyks 2006; Mejias et al. 2010) or to external fluctuating inputs (Kepecs and Raghavachari 2007; Hughes et al. 2002; Mattia and Sanchez-Vives 2012). Down state duration in these scenarios is highly variable.

Similarly, the extinction of up states has also been modeled through either deterministic or stochastic processes. Mechanisms relying on activity-dependent negative feedback bring the up state deterministically to the point of sudden transition to the down state (Fig. 1b, d). Such negative feedback can be due to activity-dependent potassium currents (Compte et al. 2003; Hill and Tononi 2005), short-term synaptic depression (Holcman and Tsodyks 2006), short-term synaptic facilitation to interneurons (Melamed et al. 2008), or slow GABAB feedback inhibition (Parga and Abbott 2007). On the other hand, endogenous fluctuations can also induce transitions from the up to the down state (Fig. 1a, c), in the form of stochastic inputs nonlinearly gated by NMDA receptors (Kepecs and Raghavachari 2007) or as a result of the stochastic dynamics of the firing rate in small populations (Mattia and Sanchez-Vives 2012). Note, however, that these recurrent networks impose complex interdependencies between all these mechanisms: as an example, short-term depression promotes up-to-down transitions on its own (Holcman and Tsodyks 2006), but prevents them when combined with adaptation currents (Benita et al. 2012).
Fig. 1

Mechanistic scenarios for up and down state switching in a bistable recurrent network with adaptation currents. (a) For fixed values of the adaptation current, the recurrent network stabilizes at firing rates that trace a cubic nullcline with a stable upper branch (up states) and a stable lower branch (down states). Unstable solutions are marked with a dashed line. In a range of values of adaptation (bistable range), the system features bistability between the two states. For very weak or absent adaptation dynamics, the systems sit at a fixed value of adaptation current (gray line) giving rise to two stable states. In this scenario, transitions between states can only occur if external fluctuations arrive that impose a change of state (wiggled arrows). (b) For strong adaptation, the nullcline defining the values of adaptation current that are steadily maintained at a given neuronal firing rate (gray line) does not intersect the network stability nullcline (black cubic line), and the system cycles periodically between up and down states. (c) Weaker adaptation makes the adaptation nullcline intersect the network nullcline in the upper branch and up states become stable. Only strong fluctuations will induce an up-to-down transition. (d) Starting from the condition in b, a slight hyperpolarization to all neurons in the network makes the down state stable, and down-to-up transitions now depend on strong episodic fluctuations

All the models proposed share a common mechanistic substrate: up and down dynamics emerge from an underlying bistability between these states that gets perturbed by additional slow or episodic mechanisms. This general dynamic picture can be illustrated graphically for a network with recurrent excitation and slow rate adaptation mechanisms (Fig. 1; Latham et al. 2000), from which it can be generalized to other mechanisms. The bistability condition implies that two stable states are accessible for a range of values for the adaptation current (bistable range, Fig. 1a). Thus, for a fixed level of adaptation (gray line in Fig. 1a), only random occasional fluctuations in excitability can impose a transition between the stable states (Fig. 1a). Adaptation mechanisms, however, can modify the system so that it no longer has stable fixed points but exhibits an oscillatory behavior. During this oscillation the network shows slow drifts along the up and down branches as adaptation builds up and decays, respectively (Fig. 1b). Depending on the magnitude of adaptation and the level of external drive to the network, transitions may occur deterministically due to the slow dynamics of adaptation (Fig. 1b, down-to-up in Fig. 1c and up-to-down in Fig. 1d), or they may be induced by episodic fluctuations that trigger transitions to escape from a stable state (Fig. 1a, up-to-down in Fig. 1c and down-to-up in Fig. 1d).

The mechanism by which bistability is achieved in the cortical network is generally assumed to be excitatory synaptic reverberation within the local circuit (Timofeev et al. 2000; Compte et al. 2003; Bazhenov et al. 2002; Hill and Tononi 2005; Holcman and Tsodyks 2006; Melamed et al. 2008; Mejias et al. 2010; Mattia and Sanchez-Vives 2012). Indeed, the bistable range of the cubic nullcline in Fig. 1 grows larger with stronger recurrent connections. This is consistent with the strong and dense connectivity in cortical circuits and explains parsimoniously the sensitivity of up states to excitatory synaptic blockers and the resilience of up-and-down membrane voltage dynamics to the hyperpolarization of individual neurons. Alternative mechanisms for such bistability are nonlinearities in NMDA receptors (Kepecs and Raghavachari 2007) or interactions between intrinsic ionic channels in individual neurons (Hughes et al. 2002; Parga and Abbott 2007). Also, intrinsic mechanisms such as persistent sodium currents have been invoked to achieve and maintain reverberatory activity in the up state (Timofeev et al. 2000; Bazhenov et al. 2002; Hill and Tononi 2005).

Several mechanisms have been proposed to explain that firing rates during up states remain low despite strong excitatory feedback in the network. Here, the most common mechanism invoked is inhibitory feedback from the local microcircuitry. A strong coupling with a local inhibitory population can control firing rates while recurrent excitation maintains up states. Consistent with this, blockers of inhibitory synaptic transmission increase the firing rates in up states (Sanchez-Vives et al. 2010). It has been argued that short-term synaptic depression can also counteract strong excitation and maintain low firing rates in up states (Holcman and Tsodyks 2006).

Finally, computational models have also addressed the role of the thalamocortical loop in slow oscillations. While thalamocortical neurons can modulate cortically generated slow oscillations (Bazhenov et al. 2002; Hill and Tononi 2005), some computational studies attribute an active role of thalamocortical neurons in maintaining these dynamics. In particular, intrinsic oscillatory mechanisms in thalamocortical neurons may take part in up state initiation in cortical networks (Hughes et al. 2002).



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© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.Systems NeuroscienceIDIBAPSBarcelonaSpain