Quantitative relations between BOLD responses, cortical energetics and impulse firing across cortical depth.

The blood oxygen level-dependent (BOLD) functional magnetic resonance imaging (fMRI) signal arises as a consequence of changes in cerebral blood flow (CBF) and cerebral metabolic rate of oxygen ( CMR O 2 ) that in turn are modulated by changes in neural activity. Recent advances in imaging have achieved sub-millimetre resolution and allowed investigation of the BOLD response as a function of cortical depth. Here, we adapt our previous theory relating the BOLD signal to neural activity to produce a quantitative model that incorporates venous blood draining between cortical layers. The adjustable inputs to the model are the neural activity and a parameter governing this blood draining. A three-layer version for transient neural inputs and a multi-layer version for constant or tonic neural inputs are able to account for a variety of experimental results, including negative BOLD signals.

A model of amygdala function following plastic changes at specific synapses during extinction.

The synaptic networks in the amygdala have been the subject of intense interest in recent times, primarily because of the role of this structure in emotion. Fear and its extinction depend on the workings of these networks, with particular interest in extinction because of its potential to ameliorate adverse symptoms associated with post-traumatic stress disorder. Here we place emphasis on the extinction networks revealed by recent techniques, and on the probable plasticity properties of their synaptic connections. We use modules of neurons representing each of the principal components identified as involved in extinction. Each of these modules consists of neural networks, containing specific ratios of excitatory and specialized inhibitory neurons as well as synaptic plasticity mechanisms appropriate for the component of the amygdala they represent. While these models can produce dynamic output, here we concentrate on the equilibrium outputs and do not model the details of the plasticity mechanisms. Pavlovian fear conditioning generates a fear memory in the lateral amygdala module that leads to activation of neurons in the basal nucleus fear module but not in the basal nucleus extinction module. Extinction protocols excite infralimbic medial prefrontal cortex neurons (IL) which in turn excite so-called extinction neurons in the amygdala, leading to the release of endocannabinoids from them and an increase in efficacy of synapses formed by lateral amygdala neurons on them. The model simulations show how such a mechanism could explain experimental observations involving the role of IL as well as endocannabinoids in different temporal phases of extinction.

Cortical Network Models of Firing Rates in the Resting and Active States Predict BOLD Responses.

Measurements of blood oxygenation level dependent (BOLD) signals have produced some surprising observations. One is that their amplitude is proportional to the entire activity in a region of interest and not just the fluctuations in this activity. Another is that during sleep and anesthesia the average BOLD correlations between regions of interest decline as the activity declines. Mechanistic explanations of these phenomena are described here using a cortical network model consisting of modules with excitatory and inhibitory neurons, taken as regions of cortical interest, each receiving excitatory inputs from outside the network, taken as subcortical driving inputs in addition to extrinsic (intermodular) connections, such as provided by associational fibers. The model shows that the standard deviation of the firing rate is proportional to the mean frequency of the firing when the extrinsic connections are decreased, so that the mean BOLD signal is proportional to both as is observed experimentally. The model also shows that if these extrinsic connections are decreased or the frequency of firing reaching the network from the subcortical driving inputs is decreased, or both decline, there is a decrease in the mean firing rate in the modules accompanied by decreases in the mean BOLD correlations between the modules, consistent with the observed changes during NREM sleep and under anesthesia. Finally, the model explains why a transient increase in the BOLD signal in a cortical area, due to a transient subcortical input, gives rises to responses throughout the cortex as observed, with these responses mediated by the extrinsic (intermodular) connections.

Cortical network models of impulse firing in the resting and active states predict cortical energetics.

Measurements of the cortical metabolic rate of glucose oxidation [CMR(glc(ox))] have provided a number of interesting and, in some cases, surprising observations. One is the decline in CMR(glc(ox)) during anesthesia and non-rapid eye movement (NREM) sleep, and another, the inverse relationship between the resting-state CMR(glc(ox)) and the transient following input from the thalamus. The recent establishment of a quantitative relationship between synaptic and action potential activity on the one hand and CMR(glc(ox)) on the other allows neural network models of such activity to probe for possible mechanistic explanations of these phenomena. We have carried out such investigations using cortical models consisting of networks of modules with excitatory and inhibitory neurons, each receiving excitatory inputs from outside the network in addition to intermodular connections. Modules may be taken as regions of cortical interest, the inputs from outside the network as arising from the thalamus, and the intermodular connections as long associational fibers. The model shows that the impulse frequency of different modules can differ from each other by less than 10%, consistent with the relatively uniform CMR(glc(ox)) observed across different regions of cortex. The model also shows that, if correlations of the average impulse rate between different modules decreases, there is a concomitant decrease in the average impulse rate in the modules, consistent with the observed drop in CMR(glc(ox)) in NREM sleep and under anesthesia. The model also explains why a transient thalamic input to sensory cortex gives rise to responses with amplitudes inversely dependent on the resting-state frequency, and therefore resting-state CMR(glc(ox)).

Fiber pathway pathology, synapse loss and decline of cortical function in schizophrenia.

A quantitative cortical model is developed, based on both computational and simulation approaches, which relates measured changes in cortical activity of gray matter with changes in the integrity of longitudinal fiber pathways. The model consists of modules of up to 5,000 neurons each, 80% excitatory and 20% inhibitory, with these having different degrees of synaptic connectiveness both within a module as well as between modules. It is shown that if the inter-modular synaptic connections are reduced to zero while maintaining the intra-modular synaptic connections constant, then activity in the modules is reduced by about 50%. This agrees with experimental observations in which cortical electrical activity in a region of interest, measured using the rate of oxidative glucose metabolism (CMRglc(ox)), is reduced by about 50% when the cortical region is isolated, either by surgical means or by transient cold block. There is also a 50% decrease in measured cortical activity following inactivation of the nucleus of Meynert and the intra-laminar nuclei of the thalamus, which arise either following appropriate lesions or in sleep. This occurs in the model if the inter-modular synaptic connections require input from these nuclei in order to function. In schizophrenia there is a 24% decrease in functional anisotropy of longitudinal fasciculi accompanied by a 7% decrease in cortical activity (CMRglc(ox)).The cortical model predicts this, namely for a 24% decrease in the functioning of the inter-modular connections, either through the complete loss of 24% of axons subserving the connections or due to such a decrease in the efficacy of all the inter-modular connections, there will be about a 7% decrease in the activity of the modules. This work suggests that deterioration of longitudinal fasciculi in schizophrenia explains the loss of activity in the gray matter.

A model of neuregulin control of NMDA receptors on synaptic spines.

Neuregulin (Nrg) through its receptor ErbB4 modulates the activity of the N-Methyl-D-Aspartate (NMDA) receptor (NMDAR) at synapses. As modification of this pathway has been implicated in schizophrenia, it is of great interest to define it in precise quantitative terms. Kinetic models of the epidermal growth factor (EGF)/ErbB receptor signalling pathway describing activation, desensitization, and tyrosine phosphorylation of EGFR/ErbB followed by binding and activation of Src family kinases that is subsequently followed by phosphorylation of target proteins are available. We have adapted these to give a kinetic description of NMDAR modulation by Nrg that recapitulates the observed kinetics of autophosphorylation of the ErbB dimer as well as the modulation of the NMDAR by Src kinase, according to whether the kinases are activated or deactivated. This quantitative description of the Nrg/NMDAR pathway provides a model for experimental elucidation of what goes awry in animal models of schizophrenia.

A model of NMDA receptor control of F-actin treadmilling in synaptic spines and their growth.

Synaptic spines grow as a consequence of the formation of F-actin filaments at the spine head. The dynamics of F-actin in the spine head upon excitation of N-methy-D-aspartate (NMDA) receptors has recently been investigated experimentally, but there is no quantitative account of how these dynamic changes occur upon activation of these receptors; this we now supply. Dynamics of F-actin at the apex of lamellipodia have been investigated in detail, giving rise to the treadmilling theory of F-actin dynamics, involving catalysis by profilin, for which quantitative models are now available. Here, we adapt such a model to describe the dynamics of F-actin in the synaptic-spine head and show that it gives quantitative descriptions of this treadmilling phenomena which are well fitted by Monte Carlo simulations. Next, the means by which excitation of NMDA receptors enhances the activity of profilin through activity of the Rho small GTPase RhoA and the specific kinase ROCK is discussed. This is then used to model the NMDA receptor excitatory enhancement of profilin and so the treadmilling process of F-actin dynamics in spine growth. Such modelling provides a quantitative description of the synaptic-spine dynamics of the filamentous to globular actin ratio that is observed experimentally.

A model for Ca2+ waves in networks of glial cells incorporating both intercellular and extracellular communication pathways.

Networks of glial cells, and in particular astrocytes, are capable of sustaining calcium (Ca(2+)) waves both in vivo and in vitro. Experimentally, it has been shown that there are two separate modes of communication: the first by the passage of an agent (inositol 1,4,5-triphosphate, IP(3)) through gap junctions (GJs) joining cells; the second by the diffusion of an extracellular agent (adenosine triphosphate, ATP) that binds to receptors on the cells. In both cases, the outcome is the release of Ca(2+) from internal stores in the glial cells. These two modes of communication are not mutually exclusive, but probably work in conjunction in many cases. We present a model of a two-dimensional network of glial cells that incorporates regenerative intercellular (GJ) and extracellular (ATP) pathways. In the extreme cases of only one type of pathway, the results are in agreement with previous models. Adding an extracellular pathway to the GJ model increased the extent and duration of the Ca(2+) wave, but did not significantly change the speed of propagation. Conversely, adding GJs to the extracellular model did increase the wave speed. The model was modified to apply to the retina by extending it to include both astrocytes and Müller cells, with GJs the dominant coupling between astrocytes and ATP responsible for most of the remaining communication. It was found that both pathways are necessary to account for experimental results.

P2X7 regenerative-loop potentiation of glutamate synaptic transmission by microglia and astrocytes.

P2X7 purinergic receptors have been implicated in chronic neuropathic and neuroinflammatory pain as well as in depression. These receptors are predominantly found in the central nervous system on microglial cells and on glutamatergic nerve terminals. Here, we develop hypotheses concerning mechanisms by which transient high-frequency impulse firing in glutamatergic terminals, such as occurs in nociceptor terminals accompanying neuropathic/neuroinflammatory pain, can lead to long-lasting changes in neural network function that is mediated by surrounding glial cells. The hypothesis consists of two parts. In the first, glutamate released by low-frequency (2Hz) terminal action potentials is insufficient to generate postsynaptic action potentials, but these are generated by brief high-frequency input bursts. Glutamate released by these bursts is partly removed by transporters on the enveloping astrocyte processes and also excites AMPA receptors on these processes, which then release ATP. This ATP is partly metabolised to adenosine, which acts on presynaptic A1 receptors to inhibit glutamate release. The remaining ATP acts on the presynaptic P2X7 receptors to facilitate glutamate release by both the high-frequency burst of action potentials as well as by a continuous low-frequency (2Hz) action potential firing that occurs in the absence of a neuropathic/neuroinflammatory insult. The positive feedback of terminal glutamate release, triggering astrocyte ATP release and leading to further glutamate release through activation of P2X7 receptors, is then sufficient to allow the normal low-frequency (2Hz) action potentials to now elicit postsynaptic action potentials after the insult is removed. In the second part of this model, the high concentration of ATP derived from astrocytes at the terminal attracts microglia by chemotaxis. The P2X7 receptors on these microglia are then engaged, resulting in microglia secreting the cytokine TNFalpha. This acts on postsynaptic TNF-R1 receptors to increase the number of AMPA receptors there, thus enhancing the efficacy of synaptic transmission. The TNFalpha also acts on presynaptic TNF-R1 to increase the amount of glutamate released by each nerve terminal impulse. Experimental tests can be made of this hypothesis that P2X7 receptors on the presynaptic terminal and those on the microglia synergistically act to ensure feedback pathways that reset to a high level the efficacy of synaptic transmission, thus ensuring chronic neuropathic/neuroinflammatory pain even when the initial insult has subsided.

A quantitative model of cortical spreading depression due to purinergic and gap-junction transmission in astrocyte networks.

Spreading depression (SD), a propagating wave of electrical silence in the cortex and archicortex, involves depolarization of neurons and astrocytes for approximately 1 min, due principally to a large increase in extracellular K+. SD is accompanied by large increases in extracellular ATP and is blocked by glutamate N-methyl-D-aspartate receptor antagonists. As a principal means of transmission between astrocytes is through their release of ATP, we have investigated if a model in which SD is driven by the effects of astrocyte waves of ATP interacting with waves of glutamate release from neurons and astrocytes can give a quantitative account of experimental observations on SD. We show that the characteristics of SD and the accompanying extracellular ionic changes can be accommodated by such a model-whether astrocyte transmission is principally through the release of ATP, as in archicortex (hippocampus) and spinal cord, or via gap junctions, as in the neocortex. Furthermore, these models give quantitative accounts of the effects on the characteristics of SD of agents toxic for astrocytes and of gap-junction blockers. Finally, an additional series of critical tests of the model is suggested.

Purinergic junctional transmission and propagation of calcium waves in cultured spinal cord microglial networks.

In order to elucidate the mechanisms of purinergic transmission of calcium (Ca(2 + )) waves between microglial cells, we have employed micro-photolithographic methods to form discrete patterns of microglia that allow quantitative measurements of Ca(2 + ) wave propagation. Microglia were confined to lanes 20-100 [Formula: see text] wide and Ca(2 + ) waves propagated from a point of mechanical stimulation, with a diminution in amplitude, for about 120 [Formula: see text]. The number of cells participating in propagation also decreased over this distance. Ca(2 + ) waves could propagate across a cell-free lane from one microglia lane to another if this distance of separation was less than about 60 [Formula: see text], indicating that propagation involved diffusion of a chemical transmitter. This transmitter was identified as ATP since all Ca(2 + ) wave propagation was blocked by the purinoceptor antagonist suramin, which blocks P2Y(2) and P2Y(12) at relatively low concentrations. Antibodies to P2Y(12) showed these at very high density compared with P2Y(2), indicating a role for P2Y(12) receptors. These observations were quantitatively accounted for by a model in which the main determinants are the diffusion of ATP released from a stimulated microglial cell and differences in the dissociation constant of the purinoceptors on the microglial cells.

Purinergic junctional transmission and propagation of calcium waves in spinal cord astrocyte networks.

Micro-photolithographic methods have been employed to form discrete patterns of spinal cord astrocytes that allow quantitative measurements of Ca(2+) wave propagation. Astrocytes were confined to lanes 20-100 microm wide and Ca(2+) waves propagated from a point of mechanical stimulation or of application of adenosine triphosphate; all Ca(2+) wave propagation was blocked by simultaneous application of purinergic P2Y(1) and P2Y(2) antagonists. Stimulation of an astrocyte at one end of a lane, followed by further stimulation of this astrocyte, gave rise to Ca(2+) transients in the same astrocytes; however, if the second stimulation was applied to an astrocyte at the other end of the lane, then this gave rise to a different but overlapping set of astrocytes generating a Ca(2+) signal. Both the amplitude and velocity of the Ca(2+) wave decreased over 270 microm from the point of initiation, and thereafter remained, on average, constant with random variations for at least a further 350 microm. Also, the percentage of astrocytes that gave a Ca(2+) transient decreased with distance along lanes. All the above observations were quantitatively predicted by our recent theoretical model of purinergic junctional transmission, as was the Ca(2+) wave propagation along and between parallel lanes of astrocytes different distances apart. These observations show that a model in which the main determinants are the diffusion of adenosine triphosphates regeneratively released from a stimulated astrocyte, together with differences in the properties and density of the purinergic P2Y receptors on astrocytes, is adequate to predict a wide range of Ca(2+) wave transmission and propagation phenomena.

Correlation model for joint development of refined retinotopic map and ocular dominance columns.

We describe a modification to a standard correlation model for the development of the geniculocortical projection that relays visual input to the visual cortex. The modification is to include threshold-activation of cortical cells as opposed to linear activation and it is shown that this can account for topographic map refinement (TMR). This contrasts with other models that require cortical cells to compete for activation or for neurotrophic support. Simulations are conducted for the joint development of ocular dominance columns and TMR in normal animals and parameter variations are used to both confirm robustness and to simulate some experimental conditions.