Editing Cognitive model (section)

===Early dynamical systems===

====Associative memory====
Early work in the application of dynamical systems to cognition can be found in the model of [[Hopfield network]]s.<ref>Hopfield, J. J. (1982). [http://www.pnas.org/content/pnas/79/8/2554.full.pdf Neural networks and physical systems with emergent collective computational abilities]. PNAS, 79, 2554-2558.</ref><ref>Hopfield, J. J. (1984). [http://www.pnas.org/content/pnas/81/10/3088.full.pdf Neurons with graded response have collective computational properties like those of two-state neurons]. PNAS, 81, 3088-3092.</ref> These networks were proposed as a model for [[associative memory (psychology)|associative memory]]. They represent the neural level of [[memory]], modeling systems of around 30 neurons which can be in either an on or off state.  By letting the [[Neural network|network]] learn on its own, structure and computational properties naturally arise. Unlike previous models, “memories” can be formed and recalled by inputting a small portion of the entire memory. Time ordering of memories can also be encoded. The behavior of the system is modeled with [[Euclidean vector|vectors]] which can change values, representing different states of the system. This early model was a major step toward a dynamical systems view of human cognition, though many details had yet to be added and more phenomena accounted for.

==== Language acquisition====
By taking into account the [[Evolutionary developmental biology|evolutionary development]] of the human [[nervous system]] and the similarity of the [[brain]] to other organs, [[Jeffrey Elman|Elman]] proposed that [[language]] and cognition should be treated as a dynamical system rather than a digital symbol processor.<ref>Elman, J. L. (1995). [http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.50.7356&rep=rep1&type=pdf Language as a dynamical system]. In R.F. Port and T. van Gelder (Eds.), Mind as motion: Explorations in the Dynamics of Cognition. (pp. 195-223). Cambridge, Massachusetts: MIT Press.</ref>  Neural networks of the type Elman implemented have come to be known as [[Recurrent neural networks|Elman networks]].  Instead of treating language as a collection of static [[Lexicon|lexical]] items and [[grammar]] rules that are learned and then used according to fixed rules, the dynamical systems view defines the [[lexicon]] as regions of state space within a dynamical system. Grammar is made up of [[attractor]]s and repellers that constrain movement in the state space. This means that representations are sensitive to context, with mental representations viewed as trajectories through mental space instead of objects that are constructed and remain static. Elman networks were trained with simple sentences to represent grammar as a dynamical system. Once a basic grammar had been learned, the networks could then parse complex sentences by predicting which words would appear next according to the dynamical model.<ref>Elman, J. L. (1991). [https://link.springer.com/content/pdf/10.1007/BF00114844.pdf Distributed representations, simple recurrent networks, and grammatical structure]. Machine Learning, 7, 195-225.</ref>

====Cognitive development====
A classic developmental error has been investigated in the context of dynamical systems:<ref>Spencer, J. P., Smith, L. B., & Thelen, E. (2001). Tests of dynamical systems account of the A-not-B error: The influence of prior experience on the spatial memory abilities of two-year-olds. Child Development, 72(5), 1327-1346.</ref><ref name="Thelen">Thelen E., Schoner, G., Scheier, C., Smith, L. B. (2001). [https://static.cambridge.org/resource/id/urn:cambridge.org:id:binary:20170214114618504-0222:S0140525X01223917:S0140525X01003910a.pdf The dynamics of embodiment: A field theory of infant preservative reaching] {{Webarchive|url=https://web.archive.org/web/20180701111727/https://static.cambridge.org/resource/id/urn:cambridge.org:id:binary:20170214114618504-0222:S0140525X01223917:S0140525X01003910a.pdf |date=2018-07-01 }}. Behavioral and Brain Sciences, 24, 1-86.</ref> The [[A-not-B error]] is proposed to be not a distinct error occurring at a specific age (8 to 10 months), but a feature of a dynamic learning process that is also present in older children. Children 2 years old were found to make an error similar to the A-not-B error when searching for toys hidden in a sandbox. After observing the toy being hidden in location A and repeatedly searching for it there, the 2-year-olds were shown a toy hidden in a new location B. When they looked for the toy, they searched in locations that were biased toward location A. This suggests that there is an ongoing representation of the toy's location that changes over time. The child's past behavior influences its model of locations of the sandbox, and so an account of behavior and learning must take into account how the system of the sandbox and the child's past actions is changing over time.<ref name="Thelen" />

====Locomotion====
One proposed mechanism of a dynamical system comes from analysis of continuous-time [[recurrent neural networks]] (CTRNNs).  By focusing on the output of the neural networks rather than their states and examining fully interconnected networks, three-neuron [[central pattern generator]] (CPG) can be used to represent systems such as leg movements during walking.<ref>Chiel, H. J., Beer, R. D., & Gallagher, J. C. (1999). Evolution and analysis of model CPGs for walking. Journal of Computational Neuroscience, 7, 99-118.</ref> This CPG contains three [[motor neuron]]s to control the foot, backward swing, and forward swing effectors of the leg. Outputs of the network represent whether the foot is up or down and how much force is being applied to generate [[torque]] in the leg joint. One feature of this pattern is that neuron outputs are either [[Binary number|off or on]] most of the time. Another feature is that the states are quasi-stable, meaning that they will eventually transition to other states. A simple pattern generator circuit like this is proposed to be a building block for a dynamical system. Sets of neurons that simultaneously transition from one quasi-stable state to another are defined as a dynamic module. These modules can in theory be combined to create larger circuits that comprise a complete dynamical system. However, the details of how this combination could occur are not fully worked out.