Quantum complex network

From Wikipedia, the free encyclopedia

Being part of network science the study of quantum complex networks aims to explore the impact of complexity science and network architectures in quantum systems. According to quantum information theory it is possible to improve communication security and data transfer rates by taking advantage of quantum mechanics. In this context the study of quantum complex networks is motivated by the possibility of quantum communications being used on a massive scale in the future. In such case it is likely that quantum communication networks will acquire non trivial features as is common in existing communication networks today.

Motivation

It is theoretically possible to take an advantage of quantum mechanics to create secure and faster communications, namely, quantum key distribution is an application of quantum cryptography that allows for theoretical completely secure communications, and quantum teleportation that can be used to transfer data at higher rate than using only classic channels.

The successful quantum teleportation experiments in 1998 followed by the development of first quantum communication networks in 2004, opened the possibility of quantum communication being used in a large scale in the future. According to findings in network science the topology of the networks is, in most cases, extremely important, and the exiting large scale communication networks today tend to have non-trivial topologies and traits, like small world effect, community structure and scale free properties. The study of networks with quantum properties and complex network topologies, can help us not only to better understand such networks but also how to use the network topology to improve the efficiency of communication networks in the future.

Important concepts

Qubits

In quantum information Qubits are the equivalent to bits in classical systems. A qubit is a property that when measured only can be found to be in one of two states, that is used to transmit information. The polarization of a photon or the nuclear spin are examples of two state systems that can be used as qubits.

Entanglement

Quantum entanglement is a physical phenomenon characterized by a correlation between the quantum states of two or more particles. While entangled particle do not interact in the classical sense, the quantum state of those particle can not be described independently. Particles can be entangle in different degrees, and the maximally entangled state are the ones the maximize the entropy of entanglement. In the context of quantum communication, quantum entanglement qubits are used as a quantum channel capable of transmitting information when combined with a classical channel.

Bell measurement

Bell measurement is joint quantum-mechanical measurement of two qubits, so that after the measurement the two qubits will be maixmailly entangle.

Entanglement swapping

Entanglement swapping is a frequent strategy used in quantum networks that allows the connections in the network to change. Lets us suppose that we have 4 qubits, A B C and D, C and D belong to the same station, while A and C belong to two different stations. Qubit A is entangled with qubit C and qubit B is entangled with qubit D. By performing a bell measurement in qubits A and B, not only the qubits A and B will be entangled but it is also possible to create an entanglement state between qubit C and qubit D, despite the fact that there was never an interactions between them. Following this process the entanglement between qubits A and C, and qubits B and D will be lost. This strategy can be use to shape the connection on the network.

Network structure

While not all models for quantum complex network follow exactly the same structure, usually nodes represent a set of qubits in the same station where operation like Bell measurements and entanglement swapping can be applied. On the other hand, a link between a node ${\displaystyle i}$ and ${\displaystyle j}$ means that a qubit in node ${\displaystyle i}$ is entangled to a qubit in node ${\displaystyle j}$, but those two qubits are in different places, thus physical interactions between them are not possible. Quantum networks where the links are interaction terms instead of entanglement may also be considered but for very different purposes. 

Notation

Each node in the network is in possession of a set of qubits that can be in different states. The most convenient representation for the quantum state of the qubits is the dirac notation and represent the two state of the qubits as ${\displaystyle |0\rangle }$ and ${\displaystyle |1\rangle }$. Two particle are entangled if the joint wave function, ${\displaystyle |\psi _{ij}\rangle }$, can not be decomposed as,

${\displaystyle |\psi _{ij}\rangle =|\phi \rangle _{i}\otimes |\phi \rangle _{j},}$

where ${\displaystyle |\phi \rangle _{i}}$ represents the quantum state of the qubit at node i and ${\displaystyle |\phi \rangle _{j}}$ represents the quantum state of the qubit at node j. Another important concept is maximally entangled states. The four states (the Bell states) that maximize the entropy of entanglement can be written as

${\displaystyle |\Phi _{ij}^{+}\rangle ={\frac {1}{\sqrt {2}}}(|0\rangle _{i}\otimes |0\rangle _{j}+|1\rangle _{i}\otimes |1\rangle _{j}),}$

${\displaystyle |\Phi _{ij}^{-}\rangle ={\frac {1}{\sqrt {2}}}(|0\rangle _{i}\otimes |0\rangle _{j}-|1\rangle _{i}\otimes |1\rangle _{j}),}$

${\displaystyle |\Psi _{ij}^{+}\rangle ={\frac {1}{\sqrt {2}}}(|0\rangle _{i}\otimes |1\rangle _{j}+|1\rangle _{i}\otimes |0\rangle _{j}),}$

${\displaystyle |\Psi _{ij}^{-}\rangle ={\frac {1}{\sqrt {2}}}(|0\rangle _{i}\otimes |1\rangle _{j}-|1\rangle _{i}\otimes |0\rangle _{j}).}$

Models

Quantum random networks

The quantum random network model proposed by Perseguers et al. can be thought of as a quantum version of the Erdős–Rényi model. Instead of the typical links used to represent other complex networks, in the quantum random network model each pair of nodes is connected through a pair of entangled qubits. In this case each node contains ${\displaystyle N-1}$ quibits, one for each other node. In a quantum random network, the degree of entanglement between a pair of nodes, represented by ${\displaystyle p}$, plays a similar role to the parameter ${\displaystyle p}$ in the Erdős–Rényi model. While in the Erdős–Rényi model two nodes form a connection with probability ${\displaystyle p}$, in the context of quantum random networks ${\displaystyle p}$ means the probability of an entangled pair of qubits being successful converted to a maximally entangled state using only local operations and classical communications, called LOCC operations. We can think of maximally entangled qubits as the true links between nodes. 

Using the notation introduced previously, we can represent a pair of entangled qubits connecting the nodes ${\displaystyle i}$ and ${\displaystyle j}$, as

${\displaystyle |\psi _{ij}\rangle ={\sqrt {1-p/2}}|0\rangle _{i}\otimes |0\rangle _{j}+{\sqrt {p/2}}|1\rangle _{i}\otimes |1\rangle _{j},}$

For ${\displaystyle p=0}$ the two qubits are not entangled,

${\displaystyle |\psi _{ij}\rangle =|0\rangle _{i}\otimes |0\rangle _{j},}$

and for ${\displaystyle p=1}$ we obtain the maximally entangled state, given by

${\displaystyle |\psi _{ij}\rangle ={\sqrt {1/2}}(|0\rangle _{i}\otimes |0\rangle _{j}+|1\rangle _{i}\otimes |1\rangle _{j})}$.

For intermediate values of ${\displaystyle p}$, ${\displaystyle 0$

, any entangled state can be, with probability ${\displaystyle p}$, successfully converted to the maximally entangled state using LOCC operations.

One of the main features that distinguish this model from its classic version is the fact the in quantum random networks links are only truly established after measurements in the networks being made, and it is possible to take advantage of this fact to shape the final state of the network. Considering an initial quantum complex network with an infinite number of nodes, Perseguers et al. showed that, by doing the right measurements and entanglement swapping, it is possible to collapse the initial network to a network containing any finite subgraph, provided that ${\displaystyle p}$ scales with ${\displaystyle N}$ as,

${\displaystyle p\sim N^{Z},}$

were ${\displaystyle Z\geq -2}$. This result is contrary to what we find in classic graph theory where the type of subgraphs contained in a network is bounded by the value of ${\displaystyle z}$.

Entanglement Percolation

The goal of entanglement percolation models is to determine if a quantum network is capable of establishing a connection between two arbitrary nodes through entanglement, and to find best the strategies to create those same connections. In a model proposed by Cirac et al. and applied to complex networks by Cuquet et al., nodes are distributed in a lattice, or in a complex network, and each pair of neighbors share two pairs of entangled qubits that can be converted to a maximally entangle qubit pair with probability ${\displaystyle p}$. We can think of maximally entangled qubits as the true links between nodes. According to classic percolation theory, considering a probability ${\displaystyle p}$ of two neighbors being connected, there is a critical ${\displaystyle p}$ designed by ${\displaystyle p_{c}}$, so that if ${\displaystyle p>p_{c}}$ there is a finite probability of existing a path between two random selected node, and for ${\displaystyle p$ the probability of existing a path between two random selected nodes goes to zero. ${\displaystyle p_{c}}$ depends only on the topology of the network. A similar phenomena was found in the model proposed by Cirac et al., where the probability of forming a maximally entangled state between two random selected nodes is zero if ${\displaystyle p$ and finite if ${\displaystyle p>p_{c}}$. The main difference between classic and entangled percolation is that in quantum networks it is possible to change the links in the network, in a way changing the effective topology of the network, as a consequence ${\displaystyle p_{c}}$ will depend on the strategy used to convert partial entangle qubits to maximally connected qubits. A naive approach yields that ${\displaystyle p_{c}}$ for a quantum network is equal to ${\displaystyle p_{c}}$ for a classic network with the same topology. Nevertheless, it was shown that is possible to take advantage of quantum swapping to lower that value, both in regular lattices and complex networks.

Network theory

From Wikipedia, the free encyclopedia

A small example network with eight vertices and ten edges

 

Network theory is the study of graphs as a representation of either symmetric relations or asymmetric relations between discrete objects. In computer science and network science, network theory is a part of graph theory: a network can be defined as a graph in which nodes and/or edges have attributes (e.g. names). 

Network theory has applications in many disciplines including statistical physics, particle physics, computer science, electrical engineering, biology, economics, finance, operations research, climatology, ecology and sociology. Applications of network theory include logistical networks, the World Wide Web, Internet, gene regulatory networks, metabolic networks, social networks, epistemological networks, etc.; see List of network theory topics for more examples. 

Euler's solution of the Seven Bridges of Königsberg problem is considered to be the first true proof in the theory of networks.

Network optimization

Break down a NP-hard network optimization task into subtasks by discarding of the most irrelevant interactions in network.

 

Network problems that involve finding an optimal way of doing something are studied under the name combinatorial optimization. Examples include network flow, shortest path problem, transport problem, transshipment problem, location problem, matching problem, assignment problem, packing problem, routing problem, critical path analysis and PERT (Program Evaluation & Review Technique). In order to break a NP-hard task of network optimization down into subtasks the network is decomposed into relatively independent subnets.

Network analysis

Electric network analysis

The electric power systems analysis could be conducted using network theory from two main points of view: 

(1) an abstract perspective (i.e., as a graph consists from nodes and edges), regardless of the electric power aspects (e.g., transmission line impedances). Most of these studies focus only on the abstract structure of the power grid using node degree distribution and betweenness distribution, which introduces substantial insight regarding the vulnerability assessment of the grid. Through these types of studies, the category of the grid structure could be identified from the complex network perspective (e.g., single-scale, scale-free). This classification might help the electric power system engineers in the planning stage or while upgrading the infrastructure (e.g., add a new transmission line) to maintain a proper redundancy level in the transmission system.

(2) weighted graphs that blend an abstract understanding of complex network theories and electric power systems properties.

Social network analysis

Visualization of social network analysis

 

Social network analysis examines the structure of relationships between social entities. These entities are often persons, but may also be groups, organizations, nation states, web sites, or scholarly publications. 

Since the 1970s, the empirical study of networks has played a central role in social science, and many of the mathematical and statistical tools used for studying networks have been first developed in sociology. Amongst many other applications, social network analysis has been used to understand the diffusion of innovations, news and rumors. Similarly, it has been used to examine the spread of both diseases and health-related behaviors. It has also been applied to the study of markets, where it has been used to examine the role of trust in exchange relationships and of social mechanisms in setting prices. Similarly, it has been used to study recruitment into political movements and social organizations. It has also been used to conceptualize scientific disagreements as well as academic prestige. More recently, network analysis (and its close cousin traffic analysis) has gained a significant use in military intelligence, for uncovering insurgent networks of both hierarchical and leaderless nature.

Biological network analysis

With the recent explosion of publicly available high throughput biological data, the analysis of molecular networks has gained significant interest. The type of analysis in this context is closely related to social network analysis, but often focusing on local patterns in the network. For example, network motifs are small subgraphs that are over-represented in the network. Similarly, activity motifs are patterns in the attributes of nodes and edges in the network that are over-represented given the network structure. Using networks to analyse patterns in biological systems, such as food-webs, allows us to visualize the nature and strength of interactions between species. The analysis of biological networks with respect to diseases has led to the development of the field of network medicine. Recent examples of application of network theory in biology include applications to understanding the cell cycle. The interactions between physiological systems like brain, heart, eyes, etc. can be regarded as a physiological network.

Narrative network analysis

Narrative network of US Elections 2012

 

The automatic parsing of textual corpora has enabled the extraction of actors and their relational networks on a vast scale. The resulting narrative networks, which can contain thousands of nodes, are then analysed by using tools from Network theory to identify the key actors, the key communities or parties, and general properties such as robustness or structural stability of the overall network, or centrality of certain nodes. This automates the approach introduced by Quantitative Narrative Analysis, whereby subject-verb-object triplets are identified with pairs of actors linked by an action, or pairs formed by actor-object.

Link analysis

Link analysis is a subset of network analysis, exploring associations between objects. An example may be examining the addresses of suspects and victims, the telephone numbers they have dialed and financial transactions that they have partaken in during a given timeframe, and the familial relationships between these subjects as a part of police investigation. Link analysis here provides the crucial relationships and associations between very many objects of different types that are not apparent from isolated pieces of information. Computer-assisted or fully automatic computer-based link analysis is increasingly employed by banks and insurance agencies in fraud detection, by telecommunication operators in telecommunication network analysis, by medical sector in epidemiology and pharmacology, in law enforcement investigations, by search engines for relevance rating (and conversely by the spammers for spamdexing and by business owners for search engine optimization), and everywhere else where relationships between many objects have to be analyzed. Links are also derived from similarity of time behavior in both nodes. Examples include climate networks where the links between two locations (nodes) are determined for example, by the similarity of the rainfall or temperature fluctuations in both sites.

Network robustness

The structural robustness of networks is studied using percolation theory. When a critical fraction of nodes (or links) is removed the network becomes fragmented into small disconnected clusters. This phenomenon is called percolation, and it represents an order-disorder type of phase transition with critical exponents. Percolation theory can predict the size of the largest component (called giant component), the critical threshold and the critical exponents.

Web link analysis

Several Web search ranking algorithms use link-based centrality metrics, including Google's PageRank, Kleinberg's HITS algorithm, the CheiRank and TrustRank algorithms. Link analysis is also conducted in information science and communication science in order to understand and extract information from the structure of collections of web pages. For example, the analysis might be of the interlinking between politicians' web sites or blogs. Another use is for classifying pages according to their mention in other pages.

Centrality measures

Information about the relative importance of nodes and edges in a graph can be obtained through centrality measures, widely used in disciplines like sociology. For example, eigenvector centrality uses the eigenvectors of the adjacency matrix corresponding to a network, to determine nodes that tend to be frequently visited. Formally established measures of centrality are degree centrality, closeness centrality, betweenness centrality, eigenvector centrality, subgraph centrality and Katz centrality. The purpose or objective of analysis generally determines the type of centrality measure to be used. For example, if one is interested in dynamics on networks or the robustness of a network to node/link removal, often the dynamical importance of a node is the most relevant centrality measure.For a centrality measure based on k-core analysis see ref.

Assortative and disassortative mixing

These concepts are used to characterize the linking preferences of hubs in a network. Hubs are nodes which have a large number of links. Some hubs tend to link to other hubs while others avoid connecting to hubs and prefer to connect to nodes with low connectivity. We say a hub is assortative when it tends to connect to other hubs. A disassortative hub avoids connecting to other hubs. If hubs have connections with the expected random probabilities, they are said to be neutral. There are three methods to quantify degree correlations.

Recurrence networks

The recurrence matrix of a recurrence plot can be considered as the adjacency matrix of an undirected and unweighted network. This allows for the analysis of time series by network measures. Applications range from detection of regime changes over characterizing dynamics to synchronization analysis.

Spread

Content in a complex network can spread via two major methods: conserved spread and non-conserved spread. In conserved spread, the total amount of content that enters a complex network remains constant as it passes through. The model of conserved spread can best be represented by a pitcher containing a fixed amount of water being poured into a series of funnels connected by tubes. Here, the pitcher represents the original source and the water is the content being spread. The funnels and connecting tubing represent the nodes and the connections between nodes, respectively. As the water passes from one funnel into another, the water disappears instantly from the funnel that was previously exposed to the water. In non-conserved spread, the amount of content changes as it enters and passes through a complex network. The model of non-conserved spread can best be represented by a continuously running faucet running through a series of funnels connected by tubes. Here, the amount of water from the original source is infinite. Also, any funnels that have been exposed to the water continue to experience the water even as it passes into successive funnels. The non-conserved model is the most suitable for explaining the transmission of most infectious diseases, neural excitation, information and rumors, etc.

Interdependent networks

An interdependent network is a system of coupled networks where nodes of one or more networks depend on nodes in other networks. Such dependencies are enhanced by the developments in modern technology. Dependencies may lead to cascading failures between the networks and a relatively small failure can lead to a catastrophic breakdown of the system. Blackouts are a fascinating demonstration of the important role played by the dependencies between networks. A recent study developed a framework to study the cascading failures in an interdependent networks system.

Neural circuit

From Wikipedia, the free encyclopedia

Anatomy of a multipolar neuron

 

A neural circuit is a population of neurons interconnected by synapses to carry out a specific function when activated. Neural circuits interconnect to one another to form large scale brain networks. Biological neural networks have inspired the design of artificial neural networks.

Early study

From "Texture of the Nervous System of Man and the Vertebrates" by Santiago Ramón y Cajal. The figure illustrates the diversity of neuronal morphologies in the auditory cortex.

 

Early treatments of neural networks can be found in Herbert Spencer's Principles of Psychology, 3rd edition (1872), Theodor Meynert's Psychiatry (1884), William James' Principles of Psychology (1890), and Sigmund Freud's Project for a Scientific Psychology (composed 1895). The first rule of neuronal learning was described by Hebb in 1949, in the Hebbian theory. Thus, Hebbian pairing of pre-synaptic and post-synaptic activity can substantially alter the dynamic characteristics of the synaptic connection and therefore either facilitate or inhibit signal transmission. In 1959, the neuroscientists, Warren Sturgis McCulloch and Walter Pitts published the first works on the processing of neural networks. They showed theoretically that networks of artificial neurons could implement logical, arithmetic, and symbolic functions. Simplified models of biological neurons were set up, now usually called perceptrons or artificial neurons. These simple models accounted for neural summation (i.e., potentials at the post-synaptic membrane will summate in the cell body). Later models also provided for excitatory and inhibitory synaptic transmission.

Connections between neurons

Proposed organization of motor-semantic neural circuits for action language comprehension. Gray dots represent areas of language comprehension, creating a network for comprehending all language. The semantic circuit of the motor system, particularly the motor representation of the legs (yellow dots), is incorporated when leg-related words are comprehended. Adapted from Shebani et al. (2013)

 

The connections between neurons in the brain are much more complex than those of the artificial neurons used in the connectionist neural computing models of artificial neural networks. The basic kinds of connections between neurons are synapses, chemical and electrical synapses. 

The establishment of synapses enables the connection of neurons into millions of overlapping, and interlinking neural circuits. Neurexins are central to this process.

One principle by which neurons work is neural summation – potentials at the postsynaptic membrane will sum up in the cell body. If the depolarization of the neuron at the axon goes above threshold an action potential will occur that travels down the axon to the terminal endings to transmit a signal to other neurons. Excitatory and inhibitory synaptic transmission is realized mostly by inhibitory postsynaptic potentials (IPSPs) and excitatory postsynaptic potentials (EPSPs). 

On the electrophysiological level, there are various phenomena which alter the response characteristics of individual synapses (called synaptic plasticity) and individual neurons (intrinsic plasticity). These are often divided into short-term plasticity and long-term plasticity. Long-term synaptic plasticity is often contended to be the most likely memory substrate. Usually the term "neuroplasticity" refers to changes in the brain that are caused by activity or experience.

Connections display temporal and spatial characteristics. Temporal characteristics refer to the continuously modified activity-dependent efficacy of synaptic transmission, called spike-timing-dependent plasticity. It has been observed in several studies that the synaptic efficacy of this transmission can undergo short-term increase (called facilitation) or decrease (depression) according to the activity of the presynaptic neuron. The induction of long-term changes in synaptic efficacy, by long-term potentiation (LTP) or depression (LTD), depends strongly on the relative timing of the onset of the excitatory postsynaptic potential and the postsynaptic action potential. LTP is induced by a series of action potentials which cause a variety of biochemical responses. Eventually, the reactions cause the expression of new receptors on the cellular membranes of the postsynaptic neurons or increase the efficacy of the existing receptors through phosphorylation.

Backpropagating action potentials cannot occur because after an action potential travels down a given segment of the axon, the m gates on voltage-gated sodium channels close, thus blocking any transient opening of the h gate from causing a change in the intracellular sodium ion (Na⁺) concentration, and preventing the generation of an action potential back towards the cell body. In some cells, however, neural backpropagation does occur through the dendritic branching and may have important effects on synaptic plasticity and computation. 

A neuron in the brain requires a single signal to a neuromuscular junction to stimulate contraction of the postsynaptic muscle cell. In the spinal cord, however, at least 75 afferent neurons are required to produce firing. This picture is further complicated by variation in time constant between neurons, as some cells can experience their EPSPs over a wider period of time than others. 

While in synapses in the developing brain synaptic depression has been particularly widely observed it has been speculated that it changes to facilitation in adult brains.

Circuitry

Model of a neural circuit in the cerebellum

 

An example of a neural circuit is the trisynaptic circuit in the hippocampus. Another is the Papez circuit linking the hypothalamus to the limbic lobe. There are several neural circuits in the cortico-basal ganglia-thalamo-cortical loop. These circuits carry information between the cortex, basal ganglia, thalamus, and back to the cortex. The largest structure within the basal ganglia, the striatum, is seen as having its own internal microcircuitry.

Neural circuits in the spinal cord called central pattern generators are responsible for controlling motor instructions involved in rhythmic behaviours. Rhythmic behaviours include walking, urination, and ejaculation. The central pattern generators are made up of different groups of spinal interneurons.

There are four principal types of neural circuits that are responsible for a broad scope of neural functions. These circuits are a diverging circuit, a converging circuit, a reverberating circuit, and a parallel after-discharge circuit.

In a diverging circuit, one neuron synapses with a number of postsynaptic cells. Each of these may synapse with many more making it possible for one neuron to stimulate up to thousands of cells. This is exemplified in the way that thousands of muscle fibers can be stimulated from the initial input from a single motor neuron.

In a converging circuit, inputs from many sources are converged into one output, affecting just one neuron or a neuron pool. This type of circuit is exemplified in the respiratory center of the brainstem, which responds to a number of inputs from different sources by giving out an appropriate breathing pattern.

A reverberating circuit produces a repetitive output. In a signalling procedure from one neuron to another in a linear sequence, one of the neurons may send a signal back to initiating neuron. Each time that the first neuron fires, the other neuron further down the sequence fires again sending it back to the source. This restimulates the first neuron and also allows the path of transmission to continue to its output. A resulting repetitive pattern is the outcome that only stops if one or more of the synapses fail, or if an inhibitory feed from another source causes it to stop. This type of reverberating circuit is found in the respiratory center that sends signals to the respiratory muscles, causing inhalation. When the circuit is interrupted by an inhibitory signal the muscles relax causing exhalation. This type of circuit may play a part in epileptic seizures.

In a parallel after-discharge circuit, a neuron inputs to several chains of neurons. Each chain is made up of a different number of neurons but their signals converge onto one output neuron. Each synapse in the circuit acts to delay the signal by about 0.5 msec so that the more synapses there are will produce a longer delay to the output neuron. After the input has stopped, the output will go on firing for some time. This type of circuit does not have a feedback loop as does the reverberating circuit. Continued firing after the stimulus has stopped is called after-discharge. This circuit type is found in the reflex arcs of certain reflexes.

Study methods

Different neuroimaging techniques have been developed to investigate the activity of neural circuits and networks. The use of "brain scanners" or functional neuroimaging to investigate the structure or function of the brain is common, either as simply a way of better assessing brain injury with high resolution pictures, or by examining the relative activations of different brain areas. Such technologies may include functional magnetic resonance imaging (fMRI), brain positron emission tomography (brain PET), and computed axial tomography (CAT) scans. Functional neuroimaging uses specific brain imaging technologies to take scans from the brain, usually when a person is doing a particular task, in an attempt to understand how the activation of particular brain areas is related to the task. In functional neuroimaging, especially fMRI, which measures hemodynamic activity (using BOLD-contrast imaging) which is closely linked to neural activity, PET, and electroencephalography (EEG) is used. 

Connectionist models serve as a test platform for different hypotheses of representation, information processing, and signal transmission. Lesioning studies in such models, e.g. artificial neural networks, where parts of the nodes are deliberately destroyed to see how the network performs, can also yield important insights in the working of several cell assemblies. Similarly, simulations of dysfunctional neurotransmitters in neurological conditions (e.g., dopamine in the basal ganglia of Parkinson's patients) can yield insights into the underlying mechanisms for patterns of cognitive deficits observed in the particular patient group. Predictions from these models can be tested in patients or via pharmacological manipulations, and these studies can in turn be used to inform the models, making the process iterative.

Clinical significance

Sometimes neural circuitries can become pathological and cause problems such as in Parkinson's disease when the basal ganglia are involved. Problems in the Papez circuit can also give rise to a number of neurodegenerative disorders including Parkinson's.