Table 1.
Sequential dynamics in neural and cognitive systems.
| Phenomenon/image | Model | References | Comments |
|---|---|---|---|
| Voting paradox / Structurally stable heteroclinic cycle | Kinetic (rate) equation, Lotka–Volterra model | Krupa, 1997; Stone and Armbruster, 1999; Ashwin et al., 2003; Postlethwaite and Dawes, 2005 | J. C. Borda and the Marquis de Condorcet (De Borda, 1781; Saari, 1995) analyzed the process of plurality elections at the French Royal Academy of Sciences. They predicted the absence of a winner in a 3 step voting process (Condorcet's triangle) |
| Learning sequences | Hopfield type non-symmetric networks with time delay including spiking neuron models | Amari, 1972; Kleinfeld, 1986; Sompolinsky and Kanter, 1986; Minai and Levy, 1993; Deco and Rolls, 2005 | Networks proposed to explain the generation of rhythmic motor patterns and the recognition and recall of sequences |
| Latching dynamics | Potts network is able to hop from one discrete attractor to another under random perturbation to make a sequence | Treves, 2005; Russo et al., 2008; Russo and Treves, 2011; Linkerhand and Gros, 2013 | The dynamics can involve sequences of continuously latching transient states |
| Sequential memory with synaptic dynamics / Chaotic itinerancy sequences of Milnor attractors or attractor ruins | Spike-frequency-adaptation mechanism Noisy dynamical systems. Cantor coding | Tsuda, 2009 | Proposed to be involved in episodic memory and itinerant process of cognition |
| Winnerless sequential switchings along metastable states/Stable heteroclinic channel | Generalized coupled Lotka–Volterra equations | Afraimovich et al., 2004; Rabinovich et al., 2008a,b | Information processing with transient dynamics at many different description levels from simple networks to cognitive processes |
| Winnerless competitive dynamics in spiking brain networks | Random inhibitory networks of spiking neurons in the striatum | Ponzi and Wickens, 2010 | Neurons form assemblies that fire in sequential coherent episodes and display complex identity–temporal spiking patterns even when cortical excitation is constant or fluctuating noisily |
| Sequences of sequences / Hierarchical transient sequences | Recognition of sequence of sequences based on a continuous dynamical model | Kiebel et al., 2009 | Speech can be considered as a sequence of sequences and can be implemented robustly by a dynamical model based on Bayesian inference. recognition dynamics disclose inference at multiple time scales |