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90-506: Observed bursts are named by the number of discrete action potentials they are composed of: a doublet is a two-spike burst, a triplet three and a quadruplet four. Neurons that are intrinsically prone to bursting behavior are referred to as bursters and this tendency to burst may be a product of the environment or the phenotype of the cell. Neurons typically operate by firing single action potential spikes in relative isolation as discrete input postsynaptic potentials combine and drive

180-420: A Banach space , and Φ is a function. When T is taken to be the integers, it is a cascade or a map . If T is restricted to the non-negative integers we call the system a semi-cascade . A cellular automaton is a tuple ( T , M , Φ), with T a lattice such as the integers or a higher-dimensional integer grid , M is a set of functions from an integer lattice (again, with one or more dimensions) to

270-401: A monoid action of T on X . The function Φ( t , x ) is called the evolution function of the dynamical system: it associates to every point x in the set X a unique image, depending on the variable t , called the evolution parameter . X is called phase space or state space , while the variable x represents an initial state of the system. We often write if we take one of

360-425: A second messenger cascade within the cell which lower calcium influx and promote calcium efflux and buffering . As calcium concentrations decline, the period of rapid bursting ceases, and the phase of quiescence begins. When calcium levels are low, the original calcium channels will reopen, restarting the process and creating a bursting pattern. In isolation or in mathematical models bursting can be recognized since

450-484: A bird feeds a brood parasite such as a cuckoo , it is unwittingly extending its phenotype; and when genes in an orchid affect orchid bee behavior to increase pollination, or when genes in a peacock affect the copulatory decisions of peahens, again, the phenotype is being extended. Genes are, in Dawkins's view, selected by their phenotypic effects. Other biologists broadly agree that the extended phenotype concept

540-508: A dynamical system. For simple dynamical systems, knowing the trajectory is often sufficient, but most dynamical systems are too complicated to be understood in terms of individual trajectories. The difficulties arise because: Many people regard French mathematician Henri Poincaré as the founder of dynamical systems. Poincaré published two now classical monographs, "New Methods of Celestial Mechanics" (1892–1899) and "Lectures on Celestial Mechanics" (1905–1910). In them, he successfully applied

630-469: A dynamical systems-motivated definition within ergodic theory that side-steps the choice of measure and assumes the choice has been made. A simple construction (sometimes called the Krylov–Bogolyubov theorem ) shows that for a large class of systems it is always possible to construct a measure so as to make the evolution rule of the dynamical system a measure-preserving transformation. In the construction

720-522: A finite set, and Φ a (locally defined) evolution function. As such cellular automata are dynamical systems. The lattice in M represents the "space" lattice, while the one in T represents the "time" lattice. Dynamical systems are usually defined over a single independent variable, thought of as time. A more general class of systems are defined over multiple independent variables and are therefore called multidimensional systems . Such systems are useful for modeling, for example, image processing . Given

810-460: A fundamental part of chaos theory , logistic map dynamics, bifurcation theory , the self-assembly and self-organization processes, and the edge of chaos concept. The concept of a dynamical system has its origins in Newtonian mechanics . There, as in other natural sciences and engineering disciplines, the evolution rule of dynamical systems is an implicit relation that gives the state of

900-518: A gene has on its surroundings, including other organisms, as an extended phenotype, arguing that "An animal's behavior tends to maximize the survival of the genes 'for' that behavior, whether or not those genes happen to be in the body of the particular animal performing it." For instance, an organism such as a beaver modifies its environment by building a beaver dam ; this can be considered an expression of its genes , just as its incisor teeth are—which it uses to modify its environment. Similarly, when

990-632: A given measure of the state space is summed for all future points of a trajectory, assuring the invariance. Some systems have a natural measure, such as the Liouville measure in Hamiltonian systems , chosen over other invariant measures, such as the measures supported on periodic orbits of the Hamiltonian system. For chaotic dissipative systems the choice of invariant measure is technically more challenging. The measure needs to be supported on

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1080-420: A global dynamical system ( R , X , Φ) on a locally compact and Hausdorff topological space X , it is often useful to study the continuous extension Φ* of Φ to the one-point compactification X* of X . Although we lose the differential structure of the original system we can now use compactness arguments to analyze the new system ( R , X* , Φ*). In compact dynamical systems the limit set of any orbit

1170-529: A large part of the Human Genome Project . Phenomics has applications in agriculture. For instance, genomic variations such as drought and heat resistance can be identified through phenomics to create more durable GMOs. Phenomics may be a stepping stone towards personalized medicine , particularly drug therapy . Once the phenomic database has acquired enough data, a person's phenomic information can be used to select specific drugs tailored to

1260-537: A multidimensional search space with several neurobiological levels, spanning the proteome, cellular systems (e.g., signaling pathways), neural systems and cognitive and behavioural phenotypes." Plant biologists have started to explore the phenome in the study of plant physiology. In 2009, a research team demonstrated the feasibility of identifying genotype–phenotype associations using electronic health records (EHRs) linked to DNA biobanks . They called this method phenome-wide association study (PheWAS). Inspired by

1350-420: A network of bursting neurons is linked it will eventually synchronize for most types of bursting. Synchronization can also appear in circuits containing no intrinsically bursting neurons, however its appearance and stability can often be improved by including intrinsically bursting cells in the network. Since synchronization is related to plasticity and memory via Hebbian plasticity and long-term potentiation

1440-437: A number of stable and unstable attractors in phase space which represent resting states. When the system is sufficiently perturbed by input stimuli it may follow a complex return path back to the stable attractor representing an action potential. In bursting neurons, these dynamic spaces bifurcate between quiescent and bursting modes according to the dynamics of the slow system. These two bifurcations may take many forms and

1530-435: A part in any or all of the following mechanisms and may have a still more sophisticated impact on the network. Synaptic strengths between neurons follow changes that depend on spike timing and bursting. For excitatory synapses of the cortex, pairing an action potential in the pre-synaptic neuron with a burst in the post-synaptic neuron leads to long-term potentiation of the synaptic strength, while pairing an action potential in

1620-453: A particular enzyme is expressed at high levels, the organism may produce more of that enzyme and exhibit a particular trait as a result. On the other hand, if the gene is expressed at low levels, the organism may produce less of the enzyme and exhibit a different trait. Gene expression is regulated at various levels and thus each level can affect certain phenotypes, including transcriptional and post-transcriptional regulation. Changes in

1710-425: A prediction about the system's future behavior, an analytical solution of such equations or their integration over time through computer simulation is realized. The study of dynamical systems is the focus of dynamical systems theory , which has applications to a wide variety of fields such as mathematics, physics, biology , chemistry , engineering , economics , history , and medicine . Dynamical systems are

1800-685: A research program carried out by many others. Oleksandr Mykolaiovych Sharkovsky developed Sharkovsky's theorem on the periods of discrete dynamical systems in 1964. One of the implications of the theorem is that if a discrete dynamical system on the real line has a periodic point of period 3, then it must have periodic points of every other period. In the late 20th century the dynamical system perspective to partial differential equations started gaining popularity. Palestinian mechanical engineer Ali H. Nayfeh applied nonlinear dynamics in mechanical and engineering systems. His pioneering work in applied nonlinear dynamics has been influential in

1890-488: A single axon . More generally, due to short-term synaptic depression and facilitation specific synapses can be resonant for certain frequencies and thus become viable specific targets for bursting cells. When combined with burst-dependent long-term plasticity, such multiplexing can allow neurons to coordinate synaptic plasticity across hierarchical networks. Burst synchronization refers to the alignment of bursting and quiescent periods in interconnected neurons. In general, if

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1980-407: Is Labrador Retriever coloring ; while the coat color depends on many genes, it is clearly seen in the environment as yellow, black, and brown. Richard Dawkins in 1978 and then again in his 1982 book The Extended Phenotype suggested that one can regard bird nests and other built structures such as caddisfly larva cases and beaver dams as "extended phenotypes". Wilhelm Johannsen proposed

2070-404: Is infinite-dimensional . This does not assume a symplectic structure . When T is taken to be the reals, the dynamical system is called global or a flow ; and if T is restricted to the non-negative reals, then the dynamical system is a semi-flow . A discrete dynamical system , discrete-time dynamical system is a tuple ( T , M , Φ), where M is a manifold locally diffeomorphic to

2160-413: Is non-empty , compact and simply connected . A dynamical system may be defined formally as a measure-preserving transformation of a measure space , the triplet ( T , ( X , Σ, μ ), Φ). Here, T is a monoid (usually the non-negative integers), X is a set , and ( X , Σ, μ ) is a probability space , meaning that Σ is a sigma-algebra on X and μ is a finite measure on ( X , Σ). A map Φ: X → X

2250-478: Is a diffeomorphism of the manifold to itself. So, f is a "smooth" mapping of the time-domain T {\displaystyle {\mathcal {T}}} into the space of diffeomorphisms of the manifold to itself. In other terms, f ( t ) is a diffeomorphism, for every time t in the domain T {\displaystyle {\mathcal {T}}} . A real dynamical system , real-time dynamical system , continuous time dynamical system , or flow

2340-528: Is a function that describes what future states follow from the current state. Often the function is deterministic , that is, for a given time interval only one future state follows from the current state. However, some systems are stochastic , in that random events also affect the evolution of the state variables. In physics , a dynamical system is described as a "particle or ensemble of particles whose state varies over time and thus obeys differential equations involving time derivatives". In order to make

2430-449: Is a more specialized phenomenon and is believed to play a much more diverse role in neural computation . Bursts differ from tonic firing, typically associated with Poisson distributed spike times for a given average firing rate, in that bursting involves a physiological "slow subsystem" that eventually depletes as the bursting continues and then must be replenished before the cell can burst again (compare refractory period ). During

2520-399: Is a tuple ( T , M , Φ) with T an open interval in the real numbers R , M a manifold locally diffeomorphic to a Banach space , and Φ a continuous function . If Φ is continuously differentiable we say the system is a differentiable dynamical system . If the manifold M is locally diffeomorphic to R , the dynamical system is finite-dimensional ; if not, the dynamical system

2610-430: Is a vector representing the cell parameters relevant to the fast subsystem, u ˙ {\displaystyle {\dot {u}}} is a vector representing the parameters of the slow modulation subsystem, and μ ≪ 1 {\displaystyle \mu \ll 1} is the ratio of the time scales between the fast and slow subsystems. Models of neuron dynamics generally exhibit

2700-463: Is found growing in two different habitats in Sweden. One habitat is rocky, sea-side cliffs , where the plants are bushy with broad leaves and expanded inflorescences ; the other is among sand dunes where the plants grow prostrate with narrow leaves and compact inflorescences. These habitats alternate along the coast of Sweden and the habitat that the seeds of Hieracium umbellatum land in, determine

2790-461: Is larger than one, or a Fano factor of the spike count that is larger than one, because bursting leads to spike patterns that are more irregular than a Poisson process (which has a CV and Fano factor equal to unity). Alternatively, the serial correlation coefficient of the ISI sequence is positive for bursting patterns, because in this case short ISIs tend to be followed by more short ISIs (at least if

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2880-508: Is now called the ergodic theorem . Combining insights from physics on the ergodic hypothesis with measure theory , this theorem solved, at least in principle, a fundamental problem of statistical mechanics . The ergodic theorem has also had repercussions for dynamics. Stephen Smale made significant advances as well. His first contribution was the Smale horseshoe that jumpstarted significant research in dynamical systems. He also outlined

2970-402: Is possible to determine all its future positions, a collection of points known as a trajectory or orbit . Before the advent of computers , finding an orbit required sophisticated mathematical techniques and could be accomplished only for a small class of dynamical systems. Numerical methods implemented on electronic computing machines have simplified the task of determining the orbits of

3060-417: Is problematic. A proposed definition for both terms as the "physical totality of all traits of an organism or of one of its subsystems" was put forth by Mahner and Kary in 1997, who argue that although scientists tend to intuitively use these and related terms in a manner that does not impede research, the terms are not well defined and usage of the terms is not consistent. Some usages of the term suggest that

3150-478: Is relevant, but consider that its role is largely explanatory, rather than assisting in the design of experimental tests. Phenotypes are determined by an interaction of genes and the environment, but the mechanism for each gene and phenotype is different. For instance, an albino phenotype may be caused by a mutation in the gene encoding tyrosinase which is a key enzyme in melanin formation. However, exposure to UV radiation can increase melanin production, hence

3240-507: Is said to be Σ-measurable if and only if, for every σ in Σ, one has Φ − 1 σ ∈ Σ {\displaystyle \Phi ^{-1}\sigma \in \Sigma } . A map Φ is said to preserve the measure if and only if, for every σ in Σ, one has μ ( Φ − 1 σ ) = μ ( σ ) {\displaystyle \mu (\Phi ^{-1}\sigma )=\mu (\sigma )} . Combining

3330-444: Is so named because the shape of the voltage trace during a burst looks similar to a square wave due to fast transitions between the resting state attractor and the spiking limit cycle. Bursting is a very general phenomenon and is observed in many contexts in many neural systems. For this reason it is difficult to find a specific meaning or purpose for bursting and instead it plays many roles. In any given circuit observed bursts may play

3420-446: Is the domain for time – there are many choices, usually the reals or the integers, possibly restricted to be non-negative. M {\displaystyle {\mathcal {M}}} is a manifold , i.e. locally a Banach space or Euclidean space, or in the discrete case a graph . f is an evolution rule t  →  f (with t ∈ T {\displaystyle t\in {\mathcal {T}}} ) such that f

3510-412: Is the hypothesized pre-cellular stage in the evolutionary history of life on earth, in which self-replicating RNA molecules proliferated prior to the evolution of DNA and proteins. The folded three-dimensional physical structure of the first RNA molecule that possessed ribozyme activity promoting replication while avoiding destruction would have been the first phenotype, and the nucleotide sequence of

3600-424: Is then ( T , M , Φ). Some formal manipulation of the system of differential equations shown above gives a more general form of equations a dynamical system must satisfy where G : ( T × M ) M → C {\displaystyle {\mathfrak {G}}:{{(T\times M)}^{M}}\to \mathbf {C} } is a functional from the set of evolution functions to

3690-607: The attractor , but attractors have zero Lebesgue measure and the invariant measures must be singular with respect to the Lebesgue measure. A small region of phase space shrinks under time evolution. For hyperbolic dynamical systems, the Sinai–Ruelle–Bowen measures appear to be the natural choice. They are constructed on the geometrical structure of stable and unstable manifolds of the dynamical system; they behave physically under small perturbations; and they explain many of

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3780-550: The genotype–phenotype distinction in 1911 to make clear the difference between an organism's hereditary material and what that hereditary material produces. The distinction resembles that proposed by August Weismann (1834–1914), who distinguished between germ plasm (heredity) and somatic cells (the body). More recently, in The Selfish Gene (1976), Dawkins distinguished these concepts as replicators and vehicles. Despite its seemingly straightforward definition,

3870-415: The hippocampal formation , is thought to perform relaying of signals originating in the hippocampus to many other parts of the brain. In order to perform this function, it uses intrinsically bursting neurons to convert promising single stimuli into longer lasting burst patterns as a way to better focus attention on new stimuli and activate important processing circuits. Once these circuits have been activated,

3960-499: The membrane potential across the threshold . Bursting can instead occur for many reasons, but neurons can be generally grouped as exhibiting input-driven or intrinsic bursting. Most cells will exhibit bursting if they are driven by a constant, subthreshold input and particular cells which are genotypically prone to bursting (called bursters ) have complex feedback systems which will produce bursting patterns with less dependence on input and sometimes even in isolation. In each case,

4050-399: The phenotype (from Ancient Greek φαίνω ( phaínō )  'to appear, show' and τύπος ( túpos )  'mark, type') is the set of observable characteristics or traits of an organism . The term covers the organism's morphology (physical form and structure), its developmental processes, its biochemical and physiological properties, its behavior , and

4140-412: The random motion of particles in the air , and the number of fish each springtime in a lake . The most general definition unifies several concepts in mathematics such as ordinary differential equations and ergodic theory by allowing different choices of the space and how time is measured. Time can be measured by integers, by real or complex numbers or can be a more general algebraic object, losing

4230-653: The above, a map Φ is said to be a measure-preserving transformation of X , if it is a map from X to itself, it is Σ-measurable, and is measure-preserving. The triplet ( T , ( X , Σ, μ ), Φ), for such a Φ, is then defined to be a dynamical system . The map Φ embodies the time evolution of the dynamical system. Thus, for discrete dynamical systems the iterates Φ n = Φ ∘ Φ ∘ ⋯ ∘ Φ {\displaystyle \Phi ^{n}=\Phi \circ \Phi \circ \dots \circ \Phi } for every integer n are studied. For continuous dynamical systems,

4320-460: The behavior of all orbits classified. In a linear system the phase space is the N -dimensional Euclidean space, so any point in phase space can be represented by a vector with N numbers. The analysis of linear systems is possible because they satisfy a superposition principle : if u ( t ) and w ( t ) satisfy the differential equation for the vector field (but not necessarily the initial condition), then so will u ( t ) +  w ( t ). For

4410-444: The bursting event, this slow subsystem modulates the timing and intensity of the emitted spikes and is thought to be important in the computational aspects of the resulting burst pattern. There are many discovered mechanisms of slow subsystems including voltage- and Ca -gated currents and spiking interplay between dendrites and the cell body . The slow subsystem also is connected to endogenous bursting patterns in neurons, where

4500-584: The bursts consist of more than two spikes). Neuron behavior is often modeled as single-compartment, non-linear dynamical systems , where the neuron states represent physiological quantities such as membrane voltage, current flow, and the concentrations of various ions intra- and extracellularly. These models most generally take the singularly perturbed form where f {\displaystyle f} and g {\displaystyle g} are both Hodgkin–Huxley style relations, x ˙ {\displaystyle {\dot {x}}}

4590-444: The choice of bifurcation both from quiescent to bursting and bursting to quiescent can affect the behavioral aspects of the burster. The complete classification of quiescent-to-bursting and bursting-to-quiescent bifurcations leads to 16 common forms and 120 possible forms if the dimensionality of the fast subsystem is not constrained. Of the most common 16, a few are well studied. The fold/homoclinic , also called square-wave, burster

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4680-450: The concept of the phenotype has hidden subtleties. It may seem that anything dependent on the genotype is a phenotype, including molecules such as RNA and proteins . Most molecules and structures coded by the genetic material are not visible in the appearance of an organism, yet they are observable (for example by Western blotting ) and are thus part of the phenotype; human blood groups are an example. It may seem that this goes beyond

4770-908: The construction and maintenance of machines and structures that are common in daily life, such as ships , cranes , bridges , buildings , skyscrapers , jet engines , rocket engines , aircraft and spacecraft . In the most general sense, a dynamical system is a tuple ( T , X , Φ) where T is a monoid , written additively, X is a non-empty set and Φ is a function with and for any x in X : for t 1 , t 2 + t 1 ∈ I ( x ) {\displaystyle \,t_{1},\,t_{2}+t_{1}\in I(x)} and   t 2 ∈ I ( Φ ( t 1 , x ) ) {\displaystyle \ t_{2}\in I(\Phi (t_{1},x))} , where we have defined

4860-517: The context of phenotype prediction. Although a phenotype is the ensemble of observable characteristics displayed by an organism, the word phenome is sometimes used to refer to a collection of traits, while the simultaneous study of such a collection is referred to as phenomics . Phenomics is an important field of study because it can be used to figure out which genomic variants affect phenotypes which then can be used to explain things like health, disease, and evolutionary fitness. Phenomics forms

4950-483: The corresponding amino acid sequence of a gene may change the frequency of guanine - cytosine base pairs ( GC content ). These base pairs have a higher thermal stability ( melting point ) than adenine - thymine , a property that might convey, among organisms living in high-temperature environments, a selective advantage on variants enriched in GC content. Richard Dawkins described a phenotype that included all effects that

5040-421: The environment and state of the neuron can be carefully observed and modulated. When observing neurons in the wild, however, bursting may be difficult to distinguish from normal firing patterns. In order to recognize bursting patterns in these contexts statistical methods are used to determine threshold parameters. Bursting is characterized by a coefficient of variation (CV) of the interspike intervals (ISI) that

5130-415: The environment plays a role in this phenotype as well. For most complex phenotypes the precise genetic mechanism remains unknown. For instance, it is largely unclear how genes determine the shape of bones or the human ear. Gene expression plays a crucial role in determining the phenotypes of organisms. The level of gene expression can affect the phenotype of an organism. For example, if a gene that codes for

5220-439: The evolution from genotype to genome to pan-genome , a concept of exploring the relationship ultimately among pan-phenome, pan-genome , and pan- envirome was proposed in 2023. Phenotypic variation (due to underlying heritable genetic variation ) is a fundamental prerequisite for evolution by natural selection . It is the living organism as a whole that contributes (or not) to the next generation, so natural selection affects

5310-440: The false statement that a "mutation has no phenotype". Behaviors and their consequences are also phenotypes, since behaviors are observable characteristics. Behavioral phenotypes include cognitive, personality, and behavioral patterns. Some behavioral phenotypes may characterize psychiatric disorders or syndromes. A phenome is the set of all traits expressed by a cell , tissue , organ , organism , or species . The term

5400-408: The field of the complex numbers. This equation is useful when modeling mechanical systems with complicated constraints. Many of the concepts in dynamical systems can be extended to infinite-dimensional manifolds—those that are locally Banach spaces —in which case the differential equations are partial differential equations . Linear dynamical systems can be solved in terms of simple functions and

5490-406: The first self-replicating RNA molecule would have been the original genotype. Dynamical systems In mathematics , a dynamical system is a system in which a function describes the time dependence of a point in an ambient space , such as in a parametric curve . Examples include the mathematical models that describe the swinging of a clock pendulum , the flow of water in a pipe ,

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5580-602: The flow through x must be defined for all time for every element of S . More commonly there are two classes of definitions for a dynamical system: one is motivated by ordinary differential equations and is geometrical in flavor; and the other is motivated by ergodic theory and is measure theoretical in flavor. In the geometrical definition, a dynamical system is the tuple ⟨ T , M , f ⟩ {\displaystyle \langle {\mathcal {T}},{\mathcal {M}},f\rangle } . T {\displaystyle {\mathcal {T}}}

5670-440: The following: where There is no need for higher order derivatives in the equation, nor for the parameter t in v ( t , x ), because these can be eliminated by considering systems of higher dimensions. Depending on the properties of this vector field, the mechanical system is called The solution can be found using standard ODE techniques and is denoted as the evolution function already introduced above The dynamical system

5760-541: The genetic structure of a population indirectly via the contribution of phenotypes. Without phenotypic variation, there would be no evolution by natural selection. The interaction between genotype and phenotype has often been conceptualized by the following relationship: A more nuanced version of the relationship is: Genotypes often have much flexibility in the modification and expression of phenotypes; in many organisms these phenotypes are very different under varying environmental conditions. The plant Hieracium umbellatum

5850-721: The individual. Large-scale genetic screens can identify the genes or mutations that affect the phenotype of an organism. Analyzing the phenotypes of mutant genes can also aid in determining gene function. Most genetic screens have used microorganisms, in which genes can be easily deleted. For instance, nearly all genes have been deleted in E. coli and many other bacteria , but also in several eukaryotic model organisms such as baker's yeast and fission yeast . Among other discoveries, such studies have revealed lists of essential genes . More recently, large-scale phenotypic screens have also been used in animals, e.g. to study lesser understood phenotypes such as behavior . In one screen,

5940-422: The interplay with plasticity and intrinsic bursting is very important. Due to the all-or-nothing nature of action potentials, single spikes can only encode information in their interspike intervals (ISI). This is an inherently low fidelity method of transferring information as it depends on very accurate timing and is sensitive to noisy loss of signal: if just a single spike is mistimed or not properly received at

6030-626: The levels of gene expression can be influenced by a variety of factors, such as environmental conditions, genetic variations, and epigenetic modifications. These modifications can be influenced by environmental factors such as diet, stress, and exposure to toxins, and can have a significant impact on an individual's phenotype. Some phenotypes may be the result of changes in gene expression due to these factors, rather than changes in genotype. An experiment involving machine learning methods utilizing gene expressions measured from RNA sequencing found that they can contain enough signal to separate individuals in

6120-513: The map Φ is understood to be a finite time evolution map and the construction is more complicated. The measure theoretical definition assumes the existence of a measure-preserving transformation. Many different invariant measures can be associated to any one evolution rule. If the dynamical system is given by a system of differential equations the appropriate measure must be determined. This makes it difficult to develop ergodic theory starting from differential equations, so it becomes convenient to have

6210-408: The memory of its physical origin, and the space may be a manifold or simply a set , without the need of a smooth space-time structure defined on it. At any given time, a dynamical system has a state representing a point in an appropriate state space . This state is often given by a tuple of real numbers or by a vector in a geometrical manifold. The evolution rule of the dynamical system

6300-619: The number of putative mutants (see table for details). Putative mutants are then tested for heritability in order to help determine the inheritance pattern as well as map out the mutations. Once they have been mapped out, cloned, and identified, it can be determined whether a mutation represents a new gene or not. These experiments showed that mutations in the rhodopsin gene affected vision and can even cause retinal degeneration in mice. The same amino acid change causes human familial blindness , showing how phenotyping in animals can inform medical diagnostics and possibly therapy. The RNA world

6390-464: The observed statistics of hyperbolic systems. The concept of evolution in time is central to the theory of dynamical systems as seen in the previous sections: the basic reason for this fact is that the starting motivation of the theory was the study of time behavior of classical mechanical systems . But a system of ordinary differential equations must be solved before it becomes a dynamic system. For example, consider an initial value problem such as

6480-411: The original intentions of the concept with its focus on the (living) organism in itself. Either way, the term phenotype includes inherent traits or characteristics that are observable or traits that can be made visible by some technical procedure. The term "phenotype" has sometimes been incorrectly used as a shorthand for the phenotypic difference between a mutant and its wild type , which would lead to

6570-399: The pattern can be maintained completely by internal mechanism without any synaptic input. This process also relies on calcium channels , which depolarize the neuron by allowing an influx of calcium ions . So long as internal calcium ion concentrations remain at an elevated level, the neuron will continue to undergo periods of rapid spiking. However, elevated calcium ion levels also trigger

6660-689: The persistent Na current is the burst initiator and the SK K current is the burst terminator. Purkinje neurons may utilise these bursting forms in information coding to the deep cerebellar nuclei . Rinzel J. (1986) A formal Classification of Bursting Mechanisms in Excitable Systems. Proceedings of the International Congress of Mathematicians. Berkeley, California, USA Izhikevich E. M. (2006) Bursting . Scholarpedia , 1(3):1300 Phenotype In genetics ,

6750-438: The phenome of a given organism is best understood as a kind of matrix of data representing physical manifestation of phenotype. For example, discussions led by A. Varki among those who had used the term up to 2003 suggested the following definition: "The body of information describing an organism's phenotypes, under the influences of genetic and environmental factors". Another team of researchers characterize "the human phenome [as]

6840-531: The phenotype that grows. An example of random variation in Drosophila flies is the number of ommatidia , which may vary (randomly) between left and right eyes in a single individual as much as they do between different genotypes overall, or between clones raised in different environments. The concept of phenotype can be extended to variations below the level of the gene that affect an organism's fitness. For example, silent mutations that do not change

6930-429: The physiological system is often thought as being the action of two linked subsystems. The fast subsystem is responsible for each spike the neuron produces. The slow subsystem modulates the shape and intensity of these spikes before eventually triggering quiescence. Input-driven bursting often encodes the intensity of input into the bursting frequency where a neuron then acts as an integrator . Intrinsic bursting

7020-661: The pre-synaptic neuron with a single spike in the post-synaptic neuron leads to long-term depression of the synaptic strength. Such dependence of synaptic plasticity on the spike timing patterns is referred to as burst-dependent plasticity. Burst-dependent plasticity is observed with variations in multiple areas of the brain. Some neurons, sometimes called resonators , exhibit sensitivity for specific input frequencies and fire either more quickly or exclusively when stimulated at that frequency. Intrinsically bursting neurons can use this band-pass filtering effect in order to encode for specific destination neurons and multiplex signals along

7110-446: The preBötC contains a heterogeneous population of both regular spiking and intrinsically bursting neurons. Intrinsically bursting neurons are thought to make the preBötC oscillations more robust to changing frequencies and the regularity of inspiratory efforts. Cerebellar Purkinje neurons have been proposed to have two distinct bursting modes: dendritically driven, by dendritic Ca spikes , and somatically driven, wherein

7200-417: The products of behavior. An organism's phenotype results from two basic factors: the expression of an organism's genetic code (its genotype ) and the influence of environmental factors. Both factors may interact, further affecting the phenotype. When two or more clearly different phenotypes exist in the same population of a species, the species is called polymorphic . A well-documented example of polymorphism

7290-553: The results of their research to the problem of the motion of three bodies and studied in detail the behavior of solutions (frequency, stability, asymptotic, and so on). These papers included the Poincaré recurrence theorem , which states that certain systems will, after a sufficiently long but finite time, return to a state very close to the initial state. Aleksandr Lyapunov developed many important approximation methods. His methods, which he developed in 1899, make it possible to define

7380-428: The role of mutations in mice were studied in areas such as learning and memory , circadian rhythmicity , vision, responses to stress and response to psychostimulants . This experiment involved the progeny of mice treated with ENU , or N-ethyl-N-nitrosourea, which is a potent mutagen that causes point mutations . The mice were phenotypically screened for alterations in the different behavioral domains in order to find

7470-551: The set I ( x ) := { t ∈ T : ( t , x ) ∈ U } {\displaystyle I(x):=\{t\in T:(t,x)\in U\}} for any x in X . In particular, in the case that U = T × X {\displaystyle U=T\times X} we have for every x in X that I ( x ) = T {\displaystyle I(x)=T} and thus that Φ defines

7560-404: The stability of sets of ordinary differential equations. He created the modern theory of the stability of a dynamical system. In 1913, George David Birkhoff proved Poincaré's " Last Geometric Theorem ", a special case of the three-body problem , a result that made him world-famous. In 1927, he published his Dynamical Systems . Birkhoff's most durable result has been his 1931 discovery of what

7650-526: The subicular signal reverts to a single spiking mode. The pre-Bötzinger complex (preBötC) is located in ventrolateral medulla and is proposed to generate the rhythm underlying inspiratory efforts in mammals. Since the frequency that the lungs need to operate at can vary according to metabolic demand, preBötC activity is modulated over a wide range of frequencies and is able to entrain the respiratory system to meet metabolic demand. While pacemaker neurons do not necessarily require intrinsically bursting neurons

7740-416: The synapse it leads to a possibly unrecoverable loss in coding. Since intrinsic bursts are thought to be derived by a computational mechanism in the slow subsystem, each can represent a much larger amount of information in the specific shape of a single burst leading to far more robust transmission. Physiological models show that for a given input the interspike and interburst timings are much more variable than

7830-424: The system for only a short time into the future. (The relation is either a differential equation , difference equation or other time scale .) To determine the state for all future times requires iterating the relation many times—each advancing time a small step. The iteration procedure is referred to as solving the system or integrating the system . If the system can be solved, then, given an initial point, it

7920-413: The timing of the burst shape itself which also implies that timing between events is a less robust way to encode information. The expanded alphabet for communication enabled by considering burst patterns as discrete signals allows for a greater channel capacity in neuronal communications and provides a popular connection between neural coding and information theory . The subiculum , a component of

8010-496: The variables as constant. The function is called the flow through x and its graph is called the trajectory through x . The set is called the orbit through x . The orbit through x is the image of the flow through x . A subset S of the state space X is called Φ- invariant if for all x in S and all t in T Thus, in particular, if S is Φ- invariant , I ( x ) = T {\displaystyle I(x)=T} for all x in S . That is,

8100-405: Was first used by Davis in 1949, "We here propose the name phenome for the sum total of extragenic, non-autoreproductive portions of the cell, whether cytoplasmic or nuclear. The phenome would be the material basis of the phenotype, just as the genome is the material basis of the genotype ." Although phenome has been in use for many years, the distinction between the use of phenome and phenotype

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