Cognitive Neurophysics Glossary
Adaptation: A naturally occurring process of interaction between brain and envi- ronment that acts on the dynamics of the brain in the form of changing the relative strength of different synapses to form dynamically evolving spatiotemporal thought/action patterns in the brain.
Anticipation: A system anticipates upcoming stable solutions. The system spends more time near a particular phase as it approaches a critical point in its present state or phase, giving rise to an enhanced phase density that specifies the locus of the upcoming state (stabilization of desired state must occur before fluctuating/destabi- lizing existing state.
This enhances phase density and prevents random spontaneous switching). Critical slowing, the increased recovery time of a state or phase, is a predictor of upcoming phase transitions. This aspect of coordination dynamics is an anticipatory dynamical system (ADS).
Attractor: The region of the state space of a dynamical system toward which trajectories travel as time passes. As long as the parameters are unchanged, if the system passes close enough to the attractor, it will never leave that region without significant perturbation. Attractors are ordered states of high stability surrounded by instability (apparently random activity). These attractors are surrounded both tem- porally and spatially by chaotic activity in the brain. Brain cell assemblies are more tightly functionally coupled in attractor regions.
Note: The Birth of an Attractor: An order parameter, once established, has a backward effect on the activity of all the elements from which it has emerged. This is the so-called slaving process. Once the collective behavior is in a state of high stability it demonstrates hysteresis prior to change.
Note: Meanings are generated through the emergent relations between attractor neural networks. Meanings are a function of an attractor network. (Melody does not consist of tones, which are the sensory data for musical experience, but of the intervals between the tones and the relations between the intervals. So it is possible for melody to be transposed in such a way that it seems to remain the same even though every individual tone has been changed.) We classify stimuli entering the attractor neural network as meaningful if they lead the network quickly to an attractor. Otherwise, sensory input is classified as meaningless and ignored. This is the foundation of perception. Through associative learning, meanings can be attached or detached from attractors.
Note: Sensitization will correspond to an opening, habituation to a closing, of attractors. Such an opening or closing of perceptual attractors can be achieved by changing attention parameters.
Related Terms:
Attractor Depth: Relative depth indicates the stability of one attractor over another. Relative depth is measured by switching behavior. A system switches more quickly from a shallow attractor to a deep one. Depth can also be measured by dwelling time and recovery speed after perturbation.
Attractor Width: A relative marker. Refers to the width of valleys in a potential well. Attractor width indicates variability inherent in the attractor space. This differs from the width of a basin of an attractor, which is defined by the set(s) of initial conditions from which the system goes into a particular behavior. The basin for each attractor would be defined by the receptor neurons that were activated during training or perception to form the nerve cell assembly.
Barrier Height: A relative marker of stability indicates amount of push or perturbation the system needs to escape the attractor.
Basin of Attraction: Collection of all points of the state space that tend to the attractor in forward time. The basin of an attractor can also be thought of as the set of initial conditions (anchors, strategies, etc.) from which the system goes to a specific behavior. The basin for each attractor in the brain is defined by the receptor neurons activated during training to form a nerve cell assembly, the whole range of sensory inputs that separately evoke a particular perception or behavior. A basin of attraction can also be thought of as the region within which all trajectories converge on a particular attractor.
Behavioral Attractor: A behavior that is stable within an individual and to which the system returns, when perturbed, acts like a behavioral attractor. Behavioral attractors are always softly assembled (functionally coupled) from interactions between their component elements (activity-dependent synapses) and are always in open energy exchange with their surroundings (other neurons and glial cells). Changes in either components (nerve cells) or in the context (sensory environment) may influence the patterns that emerge and their stability.
Critical Fluctuation: Large fluctuations as instability is approached — fluctu- ation enhancement (may be said to anticipate an ongoing pattern change).
Destabilized: An attractor is said to be destabilized when the time to recover from perturbation increases.
Development: The evolving and dissolving of sequences of attractors and the relationships between them.
Fixed-Point Attractor: Trajectory evolves toward a fixed stable point in phase space. The activity of a dampened pendulum would trace the trajectory of a fixed-point attractor. Fixed-point attractors are common for dissipative systems.
Flattening of Attractor Well: Indicates attractor instability and variability of behavior.
Instabilities: Provide a special entry point to a system because they allow a clear distinction between one pattern of behavior and another. They demarcate/separate behavioral patterns, enabling us to identify when pattern change occurs.
Landscape: A series of potential attractor wells that evolve and dissolve spa- tiotemporally over time. If a potential well is steep and narrow, it indicates that the system has few and highly stable behavioral choices. If the potential well is steep with a flat floor, it indicates that the system has several highly stable choices, none of which are preferred. A deep attractor literally “sucks in” other competing organizations of a system — the deeper the more preferred.
Limit Cycle Attractor: Patterns that repeat in time. Collective oscillations (phase locked) of neurons form stable attractors resistant to small perturbations. Habitual patterns of thought or behavior can be thought of as limit cycle attractors. Pattern interruption breaks limit cycles.
Open/Closed Attractors: When a network of cell assemblies in the brain becomes activated, the attractor is said to be open. Open indicates an active attractor basin.
Repeller: A region of state space where the state vector is repelled. Stable states are “attracting.” Unstable states are repelling.
Searching Instability: While a system is searching, it is unstable. Instabilities offer a way to find control parameters (you know when you have a control parameter when its variation/scaling causes a qualitative change). All submo- dalities can act as control parameters.
Stability: A small change in initial conditions leads to only a small change in the trajectory. A system must lose stability during phase shifts. As you weaken functionally coupled bonds, there will be greater behavioral “variability” of the order parameter. Variability is an indicator of strength of a behavioral or per- ceptual attractor. A second index of strength/stability is resistance to perturbation (see “critical slowing”).
State Vector: A point in phase space on the trajectory indicating the current state of the system. State vector describes a single point in state space.
Strange/Chaotic Attractor: Similar to limit cycle attractor. Patterns almost but never exactly repeat. Trajectories nearly but never cross. Most brain activity derived from phase portraits of EEGs depict strange attractors in the brain. A strange attractor’s trajectories never exactly repeat twice. A strange attractor is geometrically a fractal.
Trajectory: The history of a system or a state vector. Can indicate strength- ening and weakening of an attractor. A trajectory is a solution curve of a vector field. It is the path taken by the state of a system through the state’s space as time progresses.
Asymptotic Trajectories: Represent the stable behavior that is seen once initial transient effects of perturbation have died away.
Measurements of Stability:
Barrier Height: The amount of “push” or perturbation the system needs to escape the attractor (i.e., intensity of stimulus).
Critical Slowing: Refers to the ability of the system to recover from pertur- bation as it nears critical point. Critical slowing is longer the closer the system is to instability. Can also be measured as the time needed for an unstable system (attractor) to find a stable state. The time increases the closer it approaches bifurcation. The measurement of time it takes to return to some previously observed state (local relaxation time) is an important index of stability and its loss when patterns spontaneously form (can be used to test stability of newly evolved attractors).
Dwelling Time: How long a system spends in the narrow channel of an attractor well before exiting; persistence of a given state before switching occurs (increases near fixed points). The switching rate out of or back into a perceptual or behavioral attractor is a measure of relative stability of different attractors. Residence or dwelling time is also a measure of stability. (The “swish pattern” and the “physiological snap” decrease switching time out of and into precepts and behaviors. Anchors can be used to increase residence/dwelling time.)
Phase Velocity: The speed at which a state vector is captured by a particular attractor. The time it takes for a particular state/phase to enslave the collective dynamics of a system once the state vector is released from initial conditions.
Variability: Growing instability of an attractor is detectable by increased measures of variability and behavior; variability indicates the approach of a phase transition. When a system approaches transition, the range of variabil- ity around a stable mode is greatly expanded. Variability reflects different developmental trajectories leading to the stabilization or destabilization of a pattern. If you track a course of a behavior over an extended time scale, it allows you to identify places in a trajectory where new forms appear.
Readiness to Change = High Degree of Dimensional Complexity (Vari- ability/Many Degrees of Freedom)
Bifurcation: A sudden qualitative and discontinuous change (transition) in the dynamics of a system when a certain value of a parameter is reached. A point where one of the concurring modes predominates the others by slaving the ele- mentary components of the process. The predominating mode is called the order parameter (i.e., a tornado is an example of a predominating mode of air flow which has a backward effect on the action of the air molecules that make it up. Once the tornado forms its action governs the movement of molecules originally involved in creating it). When bifurcation or phase transition occurs, it causes a qualitative
change in action mode. The time needed for a system in a stable state to find a new stable state increases the closer the bifurcation is approached (see “critical slowing”). Normally, bifurcations provide a mechanism that converts one func- tional state to another.
Hopf Bifurcation: Occurs when a steady state changes to a periodic (oscil- lating) state. A very common way for periodic oscillation to “switch on.” Hopf bifurcation is one in which one stable pattern switches to another stable pattern at a critical value of spatial orientation. Two stable solutions coexist; an exchange of stability occurs at the bifurcation point as in the example of a person who begins walking from a standing-still position after being pushed from the back.
Note: A bifurcation can also manifest as a sudden transition from limit cycle activity to a chaotic (apparently random) activity, as the value of a control parameter is slightly altered.
Period-Doubling Bifurcation: A period is the amount of time it takes for a system to return to its original state. The time it takes for a system to oscillate back to its starting point doubles at certain critical values. After several period doubling cycles, the system will show no predictable period for return to its original state. Its activity will become “apparently random.” Continuous period doubling will eventually result in chaotic activity.
Saddle-Node Bifurcation: Occurs when there are attracting (stable) and repel- ling (unstable) directions in the neural coordination dynamics.
Bistability: The coexistence or simultaneous availability of two behavioral/percep- tual attractor states, tendencies, or patterns.
Catastrophe: Sudden change in the state of a continuous system when the dynamics of the system undergo a bifurcation. Change may occur suddenly and discontinu- ously even though there has only been a small change in one of the system param- eters. The magnitude of the change in the system is out of proportion to the change in the control parameter. This type of change is evident in human behavior when in one case information (a control parameter) presented to a person makes no apparent qualitative change in his/her emotions or behaviors, yet at another time the same information creates a sudden discontinuous response whose magnitude is drastically out of proportion to the previously witnessed response.
Chaos: Happens when the future of a trajectory is computationally unpredictable because small errors in the location of the trajectory lead to exponentially larger errors at later times. A chaotic dynamical system generally must meet three condi- tions: (1) it must be sensitive to initial conditions, (2) have dense periodic points, and (3) a dense orbit. Even simple systems can be chaotic having unpredictable trajectories. At the same time it can be considered “regular” in the sense that it can be completely analyzed and understood.
Chaotic Itinerancy: In the brain migration through state space along a trajectory that is, in part, determined by successive input and, in part, by input by other parts of the brain (see “reentrant”).
Circular Causality: (Between microscopic and macroscopic processes.) Macro- scopic structure and function emerge from elementary components and in turn orga- nize/govern the microscopic elements of the system they arose from (i.e., a tornado).
Control Parameter: A parameter that when scaled leads a system to explore its collective states. Control parameters in self-organized dynamical systems are non- specific, moving the system through its collective states, but not prescribing them. Control parameters break symmetry by concentrating energy in a system and induc- ing phase transition which enables the system to explore its collective vari- ables/states. Some examples of control parameters affecting the human brain are regional blood flow (head/eye movement compresses degrees of freedom). This control parameter causes phase shifts between sensory systems and subsecondary representational systems. Other examples of control parameters are gravity, motion, breathing, diet, attention, intention, training, anchors, practice, biochemistry (hor- mones, peptides) instantiated by the elicitation of reference experiences, etc.
Coupling: Elements of a system are coupled if they influence one another.
Degeneracy: Brain wiring is so overlapping that any single function can be carried out by more than one pattern of neuronal connections and a single group of neurons can participate in more than one function.
Degrees of Freedom: When degrees of freedom are exposed by dissolution of an old pattern or attractor by perturbation, the system is allowed to explore new, more functional behaviors (degrees of opportunity; see “variability”). The brain exhibits a lower functional dimension when it is in a stable, recognizable state and greater degrees of freedom during phase transition.
Dissipative System: Refers to a system that loses or dissipates energy as a function of time (i.e., a dampened pendulum that loses energy to friction).
Dynamical System: Can be thought of as a set of possible states (its phase space or state space) plus evolution rules which determine sequences of points in that space (trajectories).
Feedback: Positive feedback amplifies while negative feedback regulates (i.e., a TV
camera pointed at its own monitor is an example of visual iterative positive feedback).
Fixed Point: A resting point or equilibrium of the system. For example, the pendu- lum of a clock always eventually stops moving, so hanging vertically downward is a fixed point of this system.
Forced Resonance: When a system is acted upon by an external, periodic driving force, its oscillations become phase locked by the oscillations of the driving force. In forced resonance, the response is greatest when the frequency of the periodic
driving force matches the natural frequency of the structure. The resulting oscilla- tions are phase locked (i.e., pacing somebody’s speech rate or movement patterns to gain rapport, two clocks with oscillating pendulums will oscillate at the same frequency via vibrations carried by the wall they hang from).
Fractal: An object or process in which patterns occurring on a small spatial or temporal scale are repeated at ever larger/smaller scales.
Fractal Dimension: The self-similarity of a fractal implies that it possesses some fundamental aspect that does not vary as a function of scale. Regardless of how much a fractal is magnified, it continues to look similar in appearance. Fractal structures found in nature (trees, mountains, coastlines, clouds, the structure of the lung, etc.) have what is called statistical self-similarity. In this case, smaller crinkles are not necessarily exact copies of larger crinkles, but have the same qualitative appearance and are the same on “average.” Strange attractors also fall into this category.
Functional Synergies: Collective functional organizations that are neuronally based are subserved by soft coupling of nerve cell assemblies that render control of complex multivariable systems. Principles of self-organizing pattern formation govern their assembly.
Hysteresis: The tendency of a system to favor its history. The temporary resistance of one stable state against the dynamics of formation of another state. Hysteresis involves the persistence of a perception, state, or pattern despite settings of the control parameter that would favor the alternative. The presence of competing patterns under gradual parametric change will favor the initially established pattern.
Intermittency: Periods of stability and predictability in the midst of random fluc- tuation. Intermittency can manifest as periods of order inside randomness or periods of randomness interrupting order.
Intermittent Dynamics: The brain’s state vector, rather than residing in attractors of a neural network, dwells for varying times near attractive states where it can switch flexibly and quickly. The probability of switching will always increase as the state vector nears category boundaries. Categories are determined by the stability of attractive states. The brain lives at the edge of instability where it can switch spontaneously among collective states. Rather than requiring an active process to destabilize and switch from one stable state to another, intermittency seems to be an inherent built-in feature of neural machinery that supports perception/behavior and the brain itself. The main mechanism of intermittency is believed to be the coalescence of stable (attracting) and unstable (repelling) directions in neural coor- dination dynamics.
Iteration: The process of feeding the solution/result of an equation/process back in as “input” into that equation or process. Fractals such as that found in a shoreline are created by the process of repeated iteration. The chaotic activity of the ocean continually subtracts elements in an iterative recursive process. Submodalities can be iterated to affect mental processes in the brain (i.e., compulsion blowout).
Monostability: The existence of a single behavioral/perceptual state.
Morphogen: Any substance thought to impair or alter positional information in a developmental morphogenetic gradient.
Morphogenesis: Spontaneous self-organizing pattern formation.
Morphogenetic Field: A position/location-dependent self-organizing field. Can develop independently without instructive influences. An important property of morphogenetic fields is that it is capable of regulation, which means that any portion of the field is capable of regenerating the whole field.
Multistability: Parallel reentrant processing circuits in the brain give rise to differ- ential perception of the same physical stimulus configuration (ambiguities set up a multistable/intermittent attractor layout).
Nonlinear: Refers to a system governed by nonlinear differential or difference equations.
Nonlinearity: The emergent properties of a system are more than the sum of its parts.
Open System: A system that is free to exchange energy with the surrounding universe.
Order Parameter: A sudden, spontaneous, macroscopic reorganization resulting from the nonlinear behavior of a system where concurring modes reach a bifurcation point and one of the modes predominates the others by slaving the elementary components of the system. An order parameter acts to compress the degrees of freedom available to the elemental components of the system. This results in spon- taneous reorganization of connectivity and pattern formation.
Period: The time it takes for a trajectory on a periodic cycle to return to its starting point.
Periodic Point: Point that lies on a periodic cycle, i.e., an orbit that returns to previous values.
Perturbation: Something that perturbs or disturbs a system.
Phase Locking: Collective oscillations of neurons form limit cycles far more stable and resistant to small perturbations than a collection of individual oscillations. Habitual patterns of thought form limit cycles.
Phase Shifts: When systems are undergoing phase shifts, their components are more loosely coupled. While systems are fluid, they are freer to seek new places in their phase space and they do so when any control parameter is changed. Larger pertur- bations are required when components are tightly coupled.
Phase Space: A term describing the state space of a system that usually implies that at least one axis is a time derivative, like velocity.
Phase Transition: An autonomous reorganization of macroscopic order emerging spontaneously from elementary interactions. Phase transition occurs when a system reaches critical fluctuation. During phase transition the system lies between or near attractors, not in them. The system becomes more fluid and flexible resulting in bifurcation. The spontaneous appearance and disappearance of attractors and the restructuring of the dynamics of a system indicate phase transition.
Reafference: A command issued by the limbic system altering all the sensory systems to prepare to respond to new information.
Reentrant: Two or more abstracting networks are working disjunctively to pro- cess the same stimuli. Reentrant circuits communicate with each other in a simultaneous/parallel fashion and continually update representations of the same sampled stimulus.
Self-Organization: New and different forms emerge spontaneously due to instabil- ities (i.e., pattern interruption). Self-organization theory demonstrates pattern for- mation and change under nonspecific parametric influences. Self-organizing systems spontaneously form and change patterns due to nonlinear interactions among the components of a system.
Sensitivity to Initial Conditions: This is a fundamental to the unpredictability found in chaotic systems (see “chaos”). It means that a small change in initial condition leads to a large change in the trajectory of a state vector as time progresses. Tiny differences become drastically magnified.
Slaving Principle: In the neighborhood of critical points the behavior of a complex system is completely governed by few collective modes, the order parameters that slave all the other modes and elementary components. (Words emerge from elemen- tary processes and, in turn, govern self-organization of those processes, i.e., pain/thirst perception.)
Note: Within a complex system, long-lasting or slowly changing events govern short-lasting or swiftly changing events.
Spontaneous Pattern Formation: Is caused by symmetry breaking. Symmetry breaking occurs when there are changes in a system’s concentration of energy. When local energy levels (concentrations) change, new forms spontaneously self-organize. (Temperature, speed, ion concentration, regional blood flow, blood volume, oxygen, etc. are all components of shifting energy concentrations in the brain. All are capable of breaking symmetry.) Intention and training are specific control parameters also capable of concentrating energy and symmetry breaking.
Stability: Stability means that a small change in initial condition leads to only a small change in the trajectory.
State Space: Also referred to as Phase Space. This is the space of points whose coordinates completely specify the model system. A way of visually representing the dynamics of a system over time. The mathematical description of a dynamical
system consists of two parts, the state and the dynamics. The state is a snapshot of the system at a given instant in time, while the dynamics are the set of rules by which the state trajectory evolves over time. The state of a system is represented by the state vector. The state vector represents a point in state space.
Stochastic Resonance: Optimum noise intensity in a system, which maximizes coherent switching. Applying an optimal level of external noise can enhance weak signals. Stochastic resonance can facilitate information transmission by amplifying weak signals as they flow through the nervous system. It is thought to be a self- generated optimization process. Noise is beneficial in probing stability of coordi- nated states and discovering new ones. Noise provides fluctuation and enhancement, which is necessary for nonequilibrium phase transitions in sensory motor behavior and learning.
Switching Time: A measure of attractor stability. The amount of time it takes to switch from one attractor/pattern to another. Switching time is always faster from a less stable to a more stable attractor.
Symmetry Breaking: Results from shifting energy concentration in a system. Gives rise to spontaneous pattern formation. Symmetry breaking is a basic requisite process for obtaining information from the external environment via the human sensory systems. Symmetry removal and symmetry breaking are both basic fea- tures of the process of perception. If two regions of the visual field are related by exchange symmetry, at first they will be relegated automatically to background. The perceptual system undergoes continuous removal of exchange symmetry in order to discern pattern.
The second step of perception is accomplished suddenly, as soon as a critical state is reached where the interiorized figure falls into coincidence with, and is captured by, one of the previously established attractors in the brain system. At this point visual thinking is formed, and the order parameter develops embracing the interior- ized figure as a whole unit.
Synergetics: A branch of theoretical physics founded by Hermann Haken. Syner- getics is the study of self-organizing systems; the cooperation of individual parts of a system resulting in the spontaneous production of macroscopic, spatial, temporal, or functional structures/patterns.