The role of striatal feedforward inhibition in the maintenance of absence seizures 

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Collective dynamics in basal ganglia-thalamo-cortical net-work

Oscillations in Parkinson’s disease

There are two prominent frequency bands in oscillations of Parkinson’s disease, one peaked within 3{8 Hz (\tremor frequency ») and another within 8{15 Hz (alpha/low-beta), observed as LFP and ring rate oscillations in GPe, GPi and STN of MPTP-treated monkeys (Bergman et al. 1994; Wichmann, Bergman, and DeLong 1994; Raz, Vaadia, and Bergman 2000; Bergman et al. 1998; Wichmann et al. 1999; Dostrovsky and Bergman 2004) and human patients (Levy et al. 2000; Hutchison et al. 1997; Hayase et al. 1998; Hurtado et al. 1999; Magnin, Morel, and Jeanmonod 2000; Levy et al. 2002b). The lower frequency varies almost proportionally to the frequency of tremor (Hutchison et al. 1997) and intermittently synchronized with upper limb tremor (Hurtado et al. 1999). Existence of two frequency components strongly suggests that there are at least two di erent (but possibly over wrapping) neural networks with di erent characteristic time constants. Indeed, STN lesioning selectively suppress only the higher frequency component (Wichmann, Bergman, and DeLong 1994). Levodopa and apomorphine administration reduces low-beta frequency band and increases tremor frequency band of LFP in STN while orphenadrine enhances beta frequency band (Priori et al. 2004; see also Brown, Oliviero, and Mazzone 2001; Levy et al. 2002a). Furthermore, the tremor oscillations of di erent limbs have low coherence and thus it was suggested these oscillatory patterns are generated in di erent circuits (Ben-Pazi et al. 2001). Therefore, it is expected that there are not only two di erent networks with di erent characteristic time constants but also at least the network responsible for the lower frequency (tremor frequency) has multiple subnetworks underlying decoupled tremor oscillations. Topographicaly organized coextensive feedback loop structure in the BG may be responsible for such decoupled subnetworks.

Task-related oscillations

The oscillations in the basal ganglia are also observed in non-pathological condition. In macaque monkeys, 10{25 Hz oscillations are observed in LFP of the striatum and these oscillations are ampli ed during the period of a saccade task (Courtemanche, Fujii, and Graybiel 2003). In rats, similar task-dependent increase in oscillations are observed in theta frequency band (7{14 Hz) during the T-maze task (DeCoteau et al. 2007). It remains unclear why the dominant frequencies in rats and monkeys are di erent (Boraud et al. 2005). However, note that low frequency oscillations < 5 Hz were observed in Courtemanche, Fujii, and Graybiel (2003) but not analyzed to avoid possible artifact from cardiovascular rhythmicity ( 2 Hz) and the delta (< 5 Hz), beta (14{22 Hz) and gamma (30{50 Hz) bands were also observed in DeCoteau et al. (2007). Focal zones in the oculomotor region of the striatum found to temporary increase and decrease synchrony during saccadic eye movements (Courtemanche, Fujii, and Graybiel 2003). Brief (90{115 ms) bursts of the beta band (13{30 Hz) activity is observed at task end, after reward and post-performance period in monkeys performing movement tasks (Feingold et al. 2015). Similar bursts of the beta oscillations occur in the motor cortex but it occur after the last movement (Feingold et al. 2015).
In rats performing cued choice task, brief beta ( 20 Hz) oscillations are observed just after signal informative to make behavioral choice but not necessary during movement (Leventhal et al. 2012). The common feature of non-pathological oscillations are that they are transient in time and more focal compared to pathological oscillations (Boraud et al. 2005).

Infraslow oscillations and resting state network

There are recently developed concepts that describes oscillations in larger spatial and temporal scales. The default mode network (DMN) is de ned as the brain areas more active than other areas which show greatest deactivation during cognitive challenges (Raichle et al. 2001; Deco, Jirsa, and McIntosh 2011). A related concept, the resting state network (RSN) is de ned as the subset of brain areas functionally connected together (Biswal et al. 1995; Deco, Jirsa, and McIntosh 2011). The observation of these networks rely on activity and its dynamics of brain areas measured as blood oxygen level-dependent (BOLD) signal from fMRI and oxygen extraction fraction from positron-emission tomography (PET).
Infraslow oscillations
Prerequisite of RSN studies is neural dynamics slower than time resolution of fMRI. Indeed, resting state infraslow oscillations (0.01{0.1 Hz; also called ultraslow or multi-second oscillations) can be found in neural signals such as EEG (Monto et al. 2008; Hiltunen et al. 2014), LFP in the cor-
tex (Leopold, Murayama, and Logothetis 2003; Nir et al. 2008; Scholvinck et al. 2010) and the putamen (Pan et al. 2013), ring rate of the neurons in the cortex (Nir et al. 2008) and the BG (Ruskin, Bergstrom, and Walters 1999; Allers and Ruskin 2002) and the membrane potential of cortical neurons (Steriade, Amzica, and Nu~nez 1993; Steriade, Nu~nez, and Amzica 1993). Inter-hemispheric correlations are found in infraslow modulations of the delta frequency (1{4 Hz) power of EEG in anesthetized rats (Lu et al. 2007) and the gamma frequency (40{80 Hz) power of LFP and ring rates in humans during awake rest and sleep (Nir et al. 2008). Correlations in infraslow timescales are also found between the theta (4{7 Hz) power of EEG signals and ring rates of STN and GP in immobilized rats (Allers and Ruskin 2002) and fMRI signals and upper gamma (40{80 Hz) and lower (2{15 Hz) power of LFP in monkeys (Scholvinck et al. 2010). In a recent study using independent component analysis of human EEG and fMRI data, it was shown that BOLD signals from several RSN are correlated with full-band EEG without extracting amplitude envelopes of fast (> 1 Hz) uctuations, i.e., infraslow oscillations of BOLD and EEG signals are directly related (Hiltunen et al. 2014). These ndings indicate that cortical and subcortical net-works are collectively involved in oscillations of various frequency ranges intermittently activated at infraslow timescales. The RSN and infraslow oscillations studies often report power-law scaling (1=f -like or scale-free) (Ward and Greenwood 2007; He 2014) in the power spectral density (PSD) of the neural signals (Stam and De Bruin 2004; Lu et al. 2007; Monto et al. 2008; Nir et al. 2008). If a PSD show power-law scaling in some frequency range, it implies that there exist no promi-nent time scale and the signal is aperiodic. Often, PSD of neural signals show bump(s) on top of power-law scaling representing a predominant oscillations and background irregular dynamics, respectively.
Default mode network
The precursor of the DMN studies is the observation of consistent deactivation accompanied with cognitive demand (Andreasen et al. 1995; Nyberg et al. 1996; Shulman et al. 1997). Using PET and fMRI, Raichle et al. (2001) showed that a subset of brain areas (midline areas within the posterior cingulate and precuneus and within the medial prefrontal cortex) decreases neural activity during goal-directed behaviors compared to awake resting state with eyes closed. Since this subset of brain areas is independent of cognitive tasks, they hypothesized that these brain areas are tonically active in the baseline state. Following studies link the DMN to the regions showing infraslow uctuations in BOLD signal (Greicius et al. 2003; Fox et al. 2005; Fransson 2005; Waites et al. 2005), rather than tonically active baseline state. It has been noted that the DMN regions overlap with the regions related to self-referential or introspective mental activity such as autobiographic memory (Gusnard et al. 2001; Buckner and Carroll 2007), stimulus independent thought (Gusnard et al. 2001; Mason et al. 2007) and self-reports about mental state (Christo et al. 2009). The phase of infraslow EEG oscillations found to be strongly correlated with stimulus detection performance and suggested to be related to activation of DMN (Monto et al. 2008). Interestingly, studies using non-human primates showed that the infraslow uctuations in DMN exists even during anesthesia (Vincent et al. 2007) and light sleep (Fukunaga et al. 2006; Horovitz et al. 2008; Picchioni et al. 2008) suggesting that activation of the DMN does not imply self-referential activity although the reverse may be true.
Resting state network
Functional connectivity measured from correlations of infraslow uctuations of BOLD signals (Biswal et al. 1995; Lowe, Mock, and Sorenson 1998; Cordes et al. 2001) has been investigated prior to DMN studies. The RSN found in functional connectivity analysis is associated with the DMN by Greicius et al. (2003) for the rst time. The spontaneous activity of DMN is shown to be anticorrelated to that of an RSN active during attention-demanding cognitive tasks (Fox et al. 2005). A careful data analysis revealed nine RSN, one of which being the DMN, consistent across subjects (Damoiseaux et al. 2006). A computational work suggested that although anatomical connectivity shapes functional connectivity, the RSN changes over time and depends on the time scale at which they are measured (Honey et al. 2007). Based on this work and using fMRI and di usion spectrum imaging to access functional and structural connectivity, it was shown that functional connectivity can emerge even when direct structural connectivity is absent but such functional connectivity is variable over scanning sessions even within subject (Honey et al. 2009).
The DMN and RSN are age-dependent (Fair et al. 2008; Damoiseaux et al. 2007) and dis-rupted in patients of neuropsychiatric diseases such as autism, schizophrenia, Alzheimer’s disease, depression and attention-de cit/hyperactivity disorder (Garrity et al. 2007; Greicius 2008; Rom-bouts et al. 2009; Buckner, Andrews-Hanna, and Schacter 2008; Broyd et al. 2009). Furthermore, resting state functional connectivity of DMN is di erent from healthy controls also in patients of Parkinson’s disease even without cognitive impairment (Wu et al. 2009; Tessitore et al. 2012). The functional connectivity (Krajcovicova et al. 2012) and the deactivation pattern (Delaveau et al. 2010) of the DMN of patients of Parkinson’s disease are restored upon levodopa administra-tion. Consistent with degeneration of dopaminergic nigrostriatal neurons in Parkinson’s disease, striatal correlations with brainstem (which includes SNr) in resting state is markedly weak (Hacker et al. 2012). Increased cortex-STN (Baudrexel et al. 2011) and cortex-striatum (Hacker et al. 2012) functional connectivity is also observed. In awake rats, infraslow oscillations in the BG are enhanced by systemic dopamine injection but STN lesion only alters GP-SNr phase relation-ship while keeping the incidence of oscillations unchanged (Ruskin, Bergstrom, and Walters 1999; Ruskin et al. 2003; Hutchison et al. 2004). It indicates that the main contribution of BG to the infraslow oscillations underlying the RSN, if any, comes from the pathways through the striatum. However, computational studies typically focus only on anatomical connectivity between cortical areas (Honey et al. 2007; Deco et al. 2009; Ghosh et al. 2008; Pinotsis et al. 2013; Stam et al. 2015; Zhou et al. 2006; Zemanova, Zhou, and Kurths 2006; Deco, Jirsa, and McIntosh 2011).

Mathematical concepts for understanding the resting state networks

Deco, Jirsa, and McIntosh (2011) reviewed computational models of RSN mainly focusing on three models based on primate cortical connectivity database CoCoMac (Kotter 2004) in which each cortical area is modeled as a chaotic oscillator (Honey et al. 2007), a neural oscillator (Ghosh et al. 2008) and a Wilson-Cowan network (Deco et al. 2009). Although they show these models exhibit the infraslow oscillations and the RSN, the mathematical mechanism underlying such complex dynamics in di erent models is not clear. Here we review several mathematical concepts which may help probing into the mathematical principle of the infraslow oscillations and the RSN, in a way mathematically imprecise but applicable to neuroscience.
Structural stability
Chaos is a type of dynamics which cannot be categorized into simple types of dynamics such as static ( xed point) or oscillatory (limit cycle or period orbit) dynamics. Chaotic dynamics are often characterized by sensitive dependency on initial condition (i.e., two systems with slightly di erent states have very di erent states after a certain amount of time) even though the state of the system does not diverge. A heavily used quantity to describe the sensitive dependency on initial condition is the Lyapunov exponent which is the expansion rate of orbits averaged over long time. The term chaos is rst used by Li and Yorke (1975) although their de nition is di erent from the Lyapunov exponent-based de nition used commonly in science (see e.g., Eckmann and Ruelle 1985). The search for practically useful and rigorous de nition of chaos is still an ongoing research topic (Hunt and Ott 2015). The studies of chaos and qualitative analysis of dynamical systems in general date back Poincare’s work on celestial mechanics (Poincare 1890; Poincare and Magini 1899). Later, abundance of chaos in natural phenomena has been recognized since Lorenz (1963) found chaotic dynamics in hydrodynamic ow which indicates di culty of very-long-range weather prediction.
Soon after chaotic phenomena were recognized, one of the important questions for math-ematicians was whether such non-trivial phenomena is robust; does a system slightly di erent from the original chaotic system has the same dynamics? If chaos is not robust under slight modi cation of the system, chaotic phenomena would not be observable in experiments since one cannot reproduce experimental setting in a precisely the same way. In other words, they were asking if chaos were relevant in natural science. The dynamics of two systems are regarded as \the same » (topologically conjugate) if the state evolved in one system is the same as the state which is mapped to the state of another system rst, evolved by the law of another system, and then mapped back to the original system. Note that this condition implies the similarity of the dynamics even in the long time limit (i.e., \attractor »). Thus, given that slight di erence the state of chaotic system is expanded to a large di erence, this is a fundamental question in the theory of qualitative behavior of dynamical systems. This question is closely related to the notion of \ ne tuning of parameter » discussed a lot in computational neuroscience but more essential since the question involves the perturbations in the space of all possible dynamical systems, not just the perturbations in the parametrized space of dynamical systems. This is a very important notion in theoretical neuroscience and \interdisciplinary physics » in general where the description of the system cannot be derived from the rst principle. Such modeling always contain inaccuracy in the de nition of the model which cannot be captured by just changing the model parameters.
The robustness of qualitative dynamics under perturbations of the system de nition is concep-tualized as structural stability by Andronov and Pontryagin (1937) and subsequent works identify the condition that is su cient (Robinson 1975a; Robinson 1975b) and necessary (Ma~ne 1987; Hayashi 1997) for a dynamical system to be structurally stable. This is the condition for a theoret-ical model in science to be a priori a \good » model because otherwise there is no guarantee that the qualitative behavior remains the same if there exist uncaptured mechanisms with seemingly negligible e ect. The structural stability condition implies that the \attractor » to be hyperbolic. Roughly speaking, it means that (1) every point in the orbit of the attractor can be decomposed into the directions of expansion and contraction and (2) such decomposition is smooth within the attractor, i.e., an expanding direction cannot become contracting at any point and vice versa. Thus, in a hyperbolic system, there is no zero Lyapunov exponents except for the one in the direction of the orbit in the case of ows (continuous dynamical systems). By contraposition, zero Lyapunov exponents (not in the direction of ow) implies non-hyperbolicity and thus violates structural stability condition. Dynamical systems with near-zero maximum Lyapunov exponent are called to be at the edge of chaos in neuroscience and its positive contributions to computa-tions has been described (Sussillo and Abbott 2009; Toyoizumi and Abbott 2011). Furthermore, near-zero Lyapunov exponents (marginal modes) in general imply slow timescales which may be relevant to infraslow oscillations. However, since the systems with zero Lyapunov exponents are not structurally stable, above mentioned mathematical results demand an explanation for a system to have near-zero Lyapunov exponents robustly.
Smale (1967) conjectured that structurally stable dynamical system is generic in the space of all dynamical systems. That is to say, it is \very rare » to nd systems without structural stability. However, Newhouse (1970) found a result against Smale’s conjecture and showed that there can be a system without structural stability and all similar systems (i.e., systems in the neighborhood of the original system in the space of dynamical systems) are again not structurally stable. Note that it does not mean in nitesimal structural perturbation of the original system does not change the qualitative behavior. Instead, it means that all the systems have behavior qualitatively di erent to all other similar systems. Interestingly, such systems can have in nite number of coexisting periodic attractors (nowadays called Newhouse’s phenomena) (Newhouse 1974; Newhouse 1979; Bonatti, D az, and Pujals 2003; Pujals 2009). Although these works points to a possibility for non-hyperbolic systems to be realized in neural systems, it is still not clear how a speci c dynamical behavior can be observed in a reproducible way. It is possible that the structural stability condition is too strict so that too many \rare systems » are captured. To nd a better notion of \structural stability », mathematicians are trying to relax the condition by using di erent de nitions of \all similar systems » (see e.g., Pugh and Peixoto 2008). Another direction would be to relax the way to compare two systems (rather than using topological conjugacy). For example, if statistics of observable quantities from two systems are similar, experimentally it would be hard to detect the di erence of them even though actual temporal evaluations are asymptotically di erent. Therefore, probabilistic approach capturing stability of distribution over attractors (natural invariant measures) may be useful (Araujo 2001).
Chaotic itinerancy
Chaotic itinerancy (Kaneko and Tsuda 2003; Tsuda 2009; Tsuda 2013; Kaneko 2015) is a type of dynamics in which low-dimensional ordered dynamics appear intermittently and spontaneously in clusters (subsets) of elements in a system showing high-dimensional more random dynamics otherwise. Thus, chaotic itinerancy may be an appropriate concept to understand mathematical principle behind infraslow uctuations and intermittent activation of synchronized dynamics within clusters (i.e., RSN). Each state of low-dimensional dynamics resembles attractor but it cannot be treated as \classical » (or geometric) attractor (Kaneko and Tsuda 2003; Milnor 2006) which requires all neighboring orbits to approach to the attractor and thus are called attractor ruin (or quasiattractor). Kaneko and Tsuda (2003) proposed to model attractor ruins as Milnor attractors (Milnor 1985) whose de nition allows some substantial amount of neighboring orbits to leave the attractor. The de nition of Milnor attractor (in a broad sense) includes classical attractor but we use Milnor attractor in a narrow sense only for non-classical attractors, following Kaneko and Tsuda (2003). In compatible with their proposal, chaotic itinerancy was found in a coupled system in which each \node » has a Milnor attractor (Tsuda and Umemura 2003). In chaotic itinerancy, attraction to low-dimensional dynamics and escape from them are balanced and lead to many near-zero Lyapunov exponents (Tsuda 2013; Kaneko 2015). It suggests that chaotic itinerancy is achieved through non-hyperbolic dynamics. Indeed, even in a low dimensional system, chaotic itinerancy-like phenomena are observed if the system is non-hyperbolic and a small amount of noise is applied (Sauer 2003). Peculiarly slow timescale dynamics in non-hyperbolic systems are characterized by slow convergence of near-zero Lyapunov exponents and natural measure (probability distribution of the states in the attractor) (Anishchenko et al. 2002; Sauer 2003). Note that slow convergence of natural measure implies slow convergence of any \observable quantities » (time averages).
Counterintuitive to its non-hyperbolic nature, chaotic itinerancy has been found to abundant in high-dimensional systems according to Kaneko and Tsuda (see e.g., Kaneko 1994; Kaneko and Tsuda 2003; Tsuda 2013; Kaneko 2015). Tsuda (2009) described ve scenarios to realize chaotic itinerancy although it seems that those explanations rely on symmetries in the system or pre-de ned non-hyperbolicity (such as Milnor attractor). One scenario is that in high dimensional systems with some kind of symmetry, Milnor attractors becomes abundant due to the di erence in scaling of volume of the phase space and of the number of clustered states with respect to the number of elements in the system. On one hand, the volume of the phase space scales exponentially. On the other hand, the order of number of clustered states is factorial due to the symmetry in the system; if one state is in an attractor, the state obtained by replacing elements following the symmetry (e.g., replacing two elements in the case of permutation symmetry) is in another attractor. Since the number of clustered states grows much faster than the volume of phase space, each clustered state becomes very close to other clustered states for it to attract all neighboring states. It was observed that such phenomena are already dominating when the number of elements is 7 2 in the case of mean- eld type interaction (globally coupled maps) whose dynamics are invariant under permutation of elements (Kaneko 2002). However, it’s not clear how such symmetry arise in heterogeneous systems such as neural circuits. One may expect that mean- eld type interactions emerge in large system limit and such permutation symmetry is natural. However, if chaotic itinerancy occurred in such system, some clusters would become synchronous once in awhile hence it would break the asynchronous assumption for the mean- eld interaction to arise. Furthermore, it was shown that whether or not spatially extensive system become e ectively hyperbolic depends on the dynamics of each coupled element even with a coupling scheme having translational symmetry (Kuptsov and Politi 2011).
Other scenarios require non-structurally stable components such as Milnor attractors hence cannot answer to the question why they exist in the rst place although these scenarios provide more exible explanations for systems without symmetry to have chaotic itinerancy. Chaotic itinerancy in one of such scenarios which requires Milnor attractors and external noise has been observed in the model of sequential retrieval of memories in asynchronous neural networks (Tsuda, Koerner, and Shimizu 1987; Tsuda 1992). Although these scenarios do not account for pre-existence of non-structurally stable components without innate symmetry in the system, those scenarios still help understanding aforementioned computational models of the RSN.
Computational models of the resting state networks
Many models of RSN have a neural system near the bifurcation at each node such as a Wilson-Cowan network with parameter near the Hopf bifurcation (Deco et al. 2009), a FitzHugh-Nagumo neuron (Ghosh et al. 2008), a network of FitzHugh-Nagumo neurons (Ghosh et al. 2008; Zhou et al. 2006; Zemanova, Zhou, and Kurths 2006). All of these models have external source of noise at each node. Interestingly, Zhou et al. (2006) reported that if the nodes are replaced with self-sustained oscillators (Van der Pol), then their network cannot produce biologically plausible RSN and all nodes become synchronous. They conclude that an important requirement for RSN modeling is that nodes are \excitable », i.e., they can be transiently activated. Indeed, simple binary stochastic nodes with excitable and activated states are shown to be able to model RSN (Deco, Senden, and Jirsa 2012; Haimovici et al. 2013; Stam et al. 2015). In the scenarios of chaotic itinerancy, the excitable node can be matched to the Milnor attractor in which small per-turbation can lead to orbits leaving the attractor, i.e., excitation. However, the mechanism based on such pre-de ned Milnor-like attractor cannot explain why such attractor arise. Another related shortcoming in those models is that internode (cortico-cortical) interactions are assumed to be of mean- eld type, i.e., average of activity of the neurons in the node. It implies that neurons in a local network at a node receive co- uctuating inputs. Once mean- eld interaction constraint is taken out, it requires synchronous activity already at the local network level. Otherwise, uctua-tions from asynchronous activities in presynaptic neurons are washed out due to the e ect of the central limit theorem.
In contrast to these models, the nodes of the model of Honey et al. (2007) are not tuned to be near bifurcation and do not receive external noise. Instead, each node is a chaotic neural net-work studied in Breakspear, Terry, and Friston (2003) exhibiting intermittency, phase synchrony, and marginal stability. In other words, each node already poses the properties of chaotic itiner-ancy. Given that the low-dimensional system without structural stability exhibits chaotic itinerancy (Sauer 2003) and that the network of non-structurally stable elements (Milnor attractors) again exhibits chaotic itinerancy (Tsuda and Umemura 2003), it is natural that the network of Honey et al. (2007) can model the RSN. In the local network at each node, neurons interact with mean- eld type connection (Breakspear, Terry, and Friston 2003) as in global coupled maps hence the sce-nario based on (permutation) symmetry comes into play. Indeed, Breakspear, Terry, and Friston (2003) showed intermittent dynamics occur due to blowout bifurcation which typically requires some kind of symmetry (see e.g., Ashwin 2006). Note that chaotic itinerancy is also observed in the network of neurons which are electrically coupled to their nearest neighborhoods (Fujii and Tsuda 2004; Tsuda et al. 2004) and such connection scheme have translational symmetry. How-ever, Breakspear, Terry, and Friston (2003) also showed that intermittent behavior still exist even when heterogeneity is introduced in neuronal parameters, indicating that their system remain not to be structurally stable. It may be because their original network with symmetry happen to be a generic dynamical system such as in the set of dynamical systems described by Newhouse (1970). The mechanisms for why such dynamical systems can arise robustly remain unclear. In summary, although the scenarios of chaotic itinerancy help understanding how RSN can be modeled in dif-ferent ways, it is still not clear how those scenarios are achieved in a heterogeneous biological system. The prominent argument as to why the system can be tuned not to be structurally stable seems to be that the brain operates at critically (Deco, Senden, and Jirsa 2012; Haimovici et al. 2013; Stam et al. 2015), the hypothesis still remains controversial (see e.g., Beggs and Timme 2012).
A recently developed computational method based on large deviation theory (a generalization of statistical physics) provides a way to quantify e ective interactions among di erent degrees of freedom, hyperbolicity of the dynamics and hidden symmetries in high-dimensional dynami-cal systems, by calculating how (co-) uctuations of Lyapunov exponents scale with simulation time (Kuptsov and Politi 2011). Such method may help understanding interactions of clusters, (non-)hyperbolicity and its origin of the computational models of the infraslow uctuations and the RSN. The computational method to calculate covariant Lyapunov vectors (i.e., the expand-ing or contracting directions corresponding to Lyapunov exponents) (Ginelli et al. 2007) can be used to nd nodes which are stabilizing or destabilizing each RSN. To connect coarse-grained low-dimensional description of node and more elaborate high-dimensional network-of-networks ap-proach and also to use these computational methods e ciently, a uni ed framework in which coarse-grained model can be systematically derived from network-of-networks model is in demand. Without such a framework, understanding how a brain can become non-structurally stable (or not) and yet exhibits reproducible features such as the RSN may be di cult.

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Absence seizure

Absence seizures are characterized by brief interruptions of conscious experience accompanied with abnormal brain oscillations (2.5{4 Hz; Crunelli and Leresche 2002) recorded as spike-and-wave discharges (SWD) in an electroencephalogram (EEG) (Gibbs, Davis, and Lennox 1935). SWD are highly synchronized across a large number of cortical areas and thalamic nuclei. Absence seizures are therefore classi ed as generalized epileptic seizures (Williams 1953; Crunelli and Leresche 2002). The frequency of oscillations varies among animal models: 2{4 Hz in monkey (David et al. 1982), 3{5 Hz in cat having received a large dose of intramuscular penicillin (Gloor and Fariello 1988), 7{11 Hz in rat models (Danober et al. 1998; Coenen and Van Luijtelaar 2003) and 6{7 Hz in mouse models (McNamara 1994). Genetic Absence Epilepsy Rats from Strasbourg (GAERS) (Danober et al. 1998) and WAG/Rij strain of rats (Coenen and Van Luijtelaar 2003) are well established genetic models of absence epilepsy. These models show not only the SWD activity but also behavioral arrest concomitant to it and thus reproduces clinical aspect of human absence seizures. Mouse models such as tottering, stargazer, mocha, and lethargic also display electro-pathophysiological and clinical characteristics of absence seizures but also accompany other non-absence clinical symptoms. Therefore, relation of pathophysiological properties of these mouse models to the absence seizures is not as clear as the rat models.
Experiments have shown that both the cortex and thalamus are necessary for the maintenance of seizures (Meeren et al. 2005; Hughes 2009). SWD are abolished by cortical lesions or, more speci cally, deactivation or infusion of anti-absence drugs to the somatosensory cortex in animal models of absence epilepsy (Avoli and Gloor 1982; Vergnes and Marescaux 1992; Manning et al. 2004; Sitnikova and Luijtelaar 2004; Gurbanova et al. 2006; Polack and Charpier 2009). Moreover, lesions in the thalamus, especially the nucleus reticularis thalami (nRT), suppress SWD in rats (Buzsaki and Bickford 1988; Vergnes and Marescaux 1992; Avanzini et al. 1993; Meeren et al. 2009). These observations gave rise to the thalamocortical theory of absence epilepsy which postulates that the interactions between the thalamus and the cortex generate absence seizures (Prince and Farrell 1969; Avoli 2012).
There is converging evidence that the initiation of seizures in rodent genetic models occurs from a speci c cortical focus (Meeren et al. 2002; Meeren et al. 2005; Polack et al. 2007; Polack et al. 2009). SWD can be initiated in patients when a convulsive drug is conveyed to the cortex by intravascular injection, whereas the drug has no e ect when conveyed to the thalamus (Bennett 1953). Moreover, in rat models of absence epilepsy, neurons in the somatosensory cortex initiate SWD since they display interictal paroxysmal oscillations that do not propagate to distant cortical and thalamic areas (Polack et al. 2007) and lead the discharges in the thalamus at the beginning of the seizures (Meeren et al. 2002; Polack et al. 2007; Polack et al. 2009). By contrast, the network that maintains absence seizures over several tens of seconds has still to be identi ed. A recent study (Polack et al. 2009) showed that suppressing the thalamic region involved in the thalamocortical loop does not suppress the SWD in the somatosensory cortex. As far as we know, there is no evidence that the thalamus and cortex are su cient to maintain seizures (Danober et al. 1998; Depaulis, David, and Charpier 2015). The thalamocortical theory of absence seizures has also been studied using computational models (Destexhe 1998; Destexhe and Sejnowski 2003; Su czynski, Kalitzin, and Lopes Da Silva 2004; Bouwman et al. 2007; Taylor et al. 2013) in which GABAB inhibition from nRT to TC (thalamocortical) neurons, cortical hyperexcitability and an increased T-type current in nRT neurons play key roles.

Table of contents :

1 Introduction 
1.1 Basal ganglia-thalamo-cortical network
1.1.1 Anatomy of basal ganglia-thalamo-cortical network
1.1.2 Closed loops in the BG-thalamo-cortical network
1.1.3 Neuronal dynamics and synaptic interactions in the basal ganglia
1.1.4 Functions of the basal ganglia
1.1.5 Dysfunctions of the basal ganglia
1.2 Collective dynamics in basal ganglia-thalamo-cortical network
1.2.1 Oscillations in Parkinson’s disease
1.2.2 Task-related oscillations
1.2.3 Infraslow oscillations and resting state network
1.2.4 Mathematical concepts for understanding the resting state networks
1.3 Absence seizure
1.3.1 Involvement of the basal ganglia in absence seizure
2 The role of striatal feedforward inhibition in the maintenance of absence seizures 
2.1 Introduction
2.2 Results
2.2.1 The BG-thalamo-cortical network model
2.2.2 Strong striatal feedforward inhibition promotes bistability of the BG-thalamocorti network dynamics
2.2.3 The competition between feedback loops in the BG-thalamo-cortical network 24
2.2.4 Strong striatal feedforward inhibition promotes bistability
2.2.5 The mechanisms for bistability and suppression of MSN activity
2.2.6 Asynchronous ring and synchronous oscillations in the BG-thalamo-cortical spiking network
2.2.7 Appropriately timed excitatory stimulation of the cortex terminates seizures
2.2.8 Bistable dynamics in the network model with the GPe included
2.3 Discussion
2.3.1 Consistency of our theory with previous experimental results
2.3.2 Comparison to the thalamocortical theory
2.3.3 Perspectives and predictions
2.4 Materials and Methods
2.4.1 The rate model
2.4.2 The spiking model
2.4.3 Analysis of the rate model
2.4.4 Simulations
3 Preliminary evidence for bistable characteristics of absence seizures 
3.1 Introduction
3.2 Results
3.3 Discussion
3.4 Materials and Methods
3.4.1 In vivo experiments from epileptic animals
4 Complex dynamics of basal ganglia-thalamo-cortical loops 
4.1 Introduction
4.2 Discrete-time approximation of basal ganglia-thalamo-cortical network
4.2.1 Reduction to one dimensional discrete system
4.2.2 Analysis of the discrete model dynamics
4.2.3 Eects of synaptic dynamics
4.2.4 Degenerate and non-degenerate rate models
4.2.5 Slow feedback in the direct pathway
4.2.6 How the striatal feedforward inhibition enhances bistability
4.2.7 How the bistability depends on other network parameters
4.3 Complex dynamics
4.4 Discussion
4.4.1 Local and global bifurcation analysis
4.4.2 How to incorporate thalamocortical bursting in the discrete model
4.4.3 Incorporating topographic organization as coupled map systems
5 Discussion 
5.1 Relation to pathological oscillations
5.1.1 Scaling of pathological oscillation frequencies
5.1.2 Scenario 1: hyperdirect feedback drives B-oscillations
5.1.3 Scenario 2: hyperdirect feedback drives SWD
5.1.4 Comparison of the two scenarios
5.1.5 Inter-species scaling of alpha and beta frequency bands
5.1.6 Are Parkinsonian oscillations multi-stable?
5.2 How to determine the oscillation driver experimentally
5.3 Functional implication of the complex dynamics
A Appendix
A.1 Delayed dierential equation and the corresponding discrete system

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