Mathematical theory of chemical synaptic transmission

Mathematical theory of chemical synaptic transmission is suggested in which the modes of operation of chemical synapses are given as consequencies of some fundamental theoretical principles presented in the form of systems of quantum and macroscopic postulates. These postulates establish transmitter transfer rules between 3 component parts — cytoplasmic, vesicular and external pools of neurotransmitter. The main features of the transfers are determined by special properties of the dividing membranes (synaptic and vesicle) which show high selectivity towards the direction of the transmitter quantum transfer. The formulation of a previously unknown effect of transmitter quantum transfer from the vesicular pool into the cytoplasmic one is introduced: it is postulated that each arriving presynaptic impulse not only releases a constant fraction of the current contents of the cytoplasmic pool into the synaptic cleft (external pool), but also realizes practically simultaneous transmitter transfer from the vesicular pool into the cytoplasmic one. Zone structure of the vesicular pool is postulated. In accordance with basic equations of the theory a nonlinear control system (dynamic synaptic modulator — DYSYM) of transmitter release from the terminal is constructed.Depending on the parameters relation two types of synapses are classified — those with rapid and slow demobilization. Analytical dependencies of the transmitter pools sizes on the stimulation frequency are introduced. By fitting the frequency dependencies to the empirical data model parameters are determined corresponding to a set of experimentally studied synaptic junctions. Different aspects of the chemical synapse behaviour under the influence of presynaptic stimulation are simulated.     

Role of temperature in quanta mechanisms of facilitation in the frog neuromuscular junction

The results of computer simulations on the Double Barrier Synapse (DBS) model are presented which quantify the relationship between the synapse parameters and the quanta transfer process. The DBS model is applicable to a variety of states of synaptic activity, and by changing the synapse parameters it is possible to simulate various conditions of quanta transmission. The influence of the bathing solution temperature change on the synaptic parameters under different conditions of transmitter release in the frog neuromuscular junction is investigated. Simulations demonstrate that several synaptic parameters, including the parameters of the presynaptic membrane, are not affected by the temperature change. It is shown that a stimulation frequency exists at which the steady-state level of facilitation during a long train of stimuli is the same for a wide range of temperatures. Received: 2 August 1996 / Accepted in revised form: 19 February 1998     

Rate of quantal transmitter release at the mammalian rod synapse.

Under scotopic conditions, the mammalian rod encodes either one photon or none within its integration time. Consequently the signal presented to its synaptic terminal is binary. The synapse has a single active zone that releases neurotransmitter quanta tonically in darkness and pauses briefly in response to a rhodopsin isomerization by a photon. We asked: what minimum tonic rate would allow the postsynaptic bipolar cell to distinguish this pause from an extra-long interval between quanta due to the stochastic timing of release? The answer required a model of the circuit that included the rod convergence onto the bipolar cell and the bipolar cell''s signal-to-noise ratio. Calculations from the model suggest that tonic release must be at least 40 quanta/s. This tonic rate is much higher than at conventional synapses where reliability is achieved by employing multiple active zones. The rod''s synaptic mechanism makes efficient use of space, which in the retina is at a premium.     

Scribble Interacts with β-Catenin to Localize Synaptic Vesicles to Synapses

An understanding of how synaptic vesicles are recruited to and maintained at presynaptic compartments is required to discern the molecular mechanisms underlying presynaptic assembly and plasticity. We have previously demonstrated that cadherin–β-catenin complexes cluster synaptic vesicles at presynaptic sites. Here we show that scribble interacts with the cadherin–β-catenin complex to coordinate vesicle localization. Scribble and β-catenin are colocalized at synapses and can be coimmunoprecipitated from neuronal lysates, indicating an interaction between scribble and β-catenin at the synapse. Using an RNA interference approach, we demonstrate that scribble is important for the clustering of synaptic vesicles at synapses. Indeed, in scribble knockdown cells, there is a diffuse distribution of synaptic vesicles along the axon, and a deficit in vesicle recycling. Despite this, synapse number and the distribution of the presynaptic active zone protein, bassoon, remain unchanged. These effects largely phenocopy those observed after ablation of β-catenin. In addition, we show that loss of β-catenin disrupts scribble localization in primary neurons but that the localization of β-catenin is not dependent on scribble. Our data supports a model by which scribble functions downstream of β-catenin to cluster synaptic vesicles at developing synapses.     

Presynaptic receptors regulating the time course of neurotransmitter release from vertebrate nerve endings

A number of different types of presynaptic receptors was revealed in central and peripheral chemical synapses activated both by main mediator and co-mediators released simultaneously. Physiological significance and mechanisms of functioning of these receptors are not clear yet. They are assumed to provide negative or positive feedback decreasing or increasing the number of neurotransmitter quanta released in response to nerve impulse and thus regulating synaptic transmission. At the same time, there is one more way of secretion process modulation associated with the changes of timing of transmitter release. This mechanism was shown to contribute to the efficiency of synaptic transmission. The role of presynaptic receptors in regulation of the kinetics of quanta release is one of the interesting questions of modern neurophysiology. This paper overviews the results obtained by the authors that demonstrate the contribution of presynaptic receptors of different types into the regulation of temporal parameters of quantal secretion at the vertebrates neuromuscular junction. It was shown that activation of the cholinergic nicotinic receptors leads to a decrease of the amplitude of postsynaptic response not only due to reduction of the quantity of released quanta but also due to increased the level of asynchronous release. On the contrary, the facilitating effect of catecholamines on the neuromuscular synapse is the result of activation of presynaptic β₁-adrenoreceptors which leads to greater synchronization of release process and, consequently, to the increase of the amplitude of the postsynaptic response. Presynaptic purine receptors, involved in the modulation the intensity of secretion, are also capable of alteration of the time course of secretion. Activation of ryanodine receptors results in the increase of the number of quanta released with prolonged latencies leading to appearance of the phase of delayed asynchronous neurotransmitter release.     

Presynaptic CK2 promotes synapse organization and stability by targeting Ankyrin2

The precise regulation of synapse maintenance is critical to the development and function of neuronal circuits. Using an in vivo RNAi screen targeting the Drosophila kinome and phosphatome, we identify 11 kinases and phosphatases controlling synapse stability by regulating cytoskeletal, phospholipid, or metabolic signaling. We focus on casein kinase 2 (CK2) and demonstrate that the regulatory (β) and catalytic (α) subunits of CK2 are essential for synapse maintenance. CK2α kinase activity is required in the presynaptic motoneuron, and its interaction with CK2β, mediated cooperatively by two N-terminal residues of CK2α, is essential for CK2 holoenzyme complex stability and function in vivo. Using genetic and biochemical approaches we identify Ankyrin2 as a key presynaptic target of CK2 to maintain synapse stability. In addition, CK2 activity controls the subcellular organization of individual synaptic release sites within the presynaptic nerve terminal. Our study identifies phosphorylation of structural synaptic components as a compelling mechanism to actively control the development and longevity of synaptic connections.     

Multiple spike initiation zones in a neuron implicated in learning in the leech: a computational model

Sensitization of the defensive shortening reflex in the leech has been linked to a segmentally repeated tri-synaptic positive feedback loop. Serotonin from the R-cell enhances S-cell excitability, S-cell impulses cross an electrical synapse into the C-interneuron, and the C-interneuron excites the R-cell via a glutamatergic synapse. The C-interneuron has two unusual characteristics. First, impulses take longer to propagate from the S soma to the C soma than in the reverse direction. Second, impulses recorded from the electrically unexcitable C soma vary in amplitude when extracellular divalent cation concentrations are elevated, with smaller impulses failing to induce synaptic potentials in the R-cell. A compartmental, computational model was developed to test the sufficiency of multiple, independent spike initiation zones in the C-interneuron to explain these observations. The model displays asymmetric delays in impulse propagation across the S–C electrical synapse and graded impulse amplitudes in the C-interneuron in simulated high divalent cation concentrations.     

Synapses are regulated by the cytoplasmic tyrosine kinase Fer in a pathway mediated by p120catenin,Fer, SHP-2, and β-catenin

Localization of presynaptic components to synaptic sites is critical for hippocampal synapse formation. Cell adhesion–regulated signaling is important for synaptic development and function, but little is known about differentiation of the presynaptic compartment. In this study, we describe a pathway that promotes presynaptic development involving p120catenin (p120ctn), the cytoplasmic tyrosine kinase Fer, the protein phosphatase SHP-2, and β-catenin. Presynaptic Fer depletion prevents localization of active zone constituents and synaptic vesicles and inhibits excitatory synapse formation and synaptic transmission. Depletion of p120ctn or SHP-2 similarly disrupts synaptic vesicle localization with active SHP-2, restoring synapse formation in the absence of Fer. Fer or SHP-2 depletion results in elevated tyrosine phosphorylation of β-catenin. β-Catenin overexpression restores normal synaptic vesicle localization in the absence of Fer or SHP-2. Our results indicate that a presynaptic signaling pathway through p120ctn, Fer, SHP-2, and β-catenin promotes excitatory synapse development and function.     

The analysis of nonlinear synaptic transmission

In order to characterize synaptic transmission at a unitary facilitating synapse in the lobster cardiac ganglion, a new nonlinear systems analysis technique for discrete-input systems was developed and applied. From the output of the postsynaptic cell in response to randomly occurring presynaptic nerve impulses, a set of kernels, analogous to Wiener kernels, was computed. The kernels up to third order served to characterize, with reasonable accuracy, the input-output properties of the synapse. A mathematical model of the synapse was also tested with a random impulse train and model predictions were compared with experimental synaptic output. Although the model proved to be even more accurate overall than the kernel characterization, there were slight but consistent errors in the model's performance. These were also reflected as differences between model and experimental kernels. It is concluded that a random train analysis provides a comprehensive and objective comparison between model and experiment and automatically provides an arbitrarily accurate characterization of a system's input-output behavior, even in complicated cases where other approaches are impractical.     

Synaptic Competition Sculpts the Development of GABAergic Axo-Dendritic but Not Perisomatic Synapses

The neurotransmitter GABA regulates many aspects of inhibitory synapse development. We tested the hypothesis that GABA_A receptors (GABA_ARs) work together with the synaptic adhesion molecule neuroligin 2 (NL2) to regulate synapse formation in different subcellular compartments. We investigated mice (“γ2 knockdown mice”) with an engineered allele of the GABA_AR γ2 subunit gene which produced a mosaic expression of synaptic GABA_ARs in neighboring neurons, causing a strong imbalance in synaptic inhibition. Deletion of the γ2 subunit did not abolish synapse formation or the targeting of NL2 to distinct types of perisomatic and axo-dendritic contacts. Thus synaptic localization of NL2 does not require synaptic GABA_ARs. However, loss of the γ2 subunit caused a selective decrease in the number of axo-dendritic synapses on cerebellar Purkinje cells and cortical pyramidal neurons, whereas perisomatic synapses were not significantly affected. Notably, γ2-positive cells had increased axo-dendritic innervation compared with both γ2-negative and wild-type counterparts. Moreover heterologous synapses on spines, that are found after total deletion of GABA_ARs from all Purkinje cells, were rare in cerebella of γ2 knockdown mice. These findings reveal a selective role of γ2 subunit-containing GABA_ARs in regulating synapse development in distinct subcellular compartments, and support the hypothesis that the refinement of axo-dendritic synapses is regulated by activity-dependent competition between neighboring neurons.     

The Analysis of Synaptically Generated Traveling Waves

Mathematical and computational models for the propagation of activity in excitatorily coupled neurons are simulated and analyzed. The basic measurable quantity—velocity—is found for a wide class of models. Numerical bifurcation techniques, asymptotic analysis, and numerical simulations are used to show that there are distinct scaling laws for the velocity as a function of a variety of parameters. In particular, the obvious linear relationships between speed and spatial spread or synaptic decay rate are shown. More surprisingly, it is shown that the velocity scales as a power law with synaptic coupling strength and that the exponent is dependent only on the rising phase of the synapse.     

Theory of ocular dominance column formation

A general theory previously proposed by the author which describes synaptic stabilization on the basis of three basic assumptions is employed for the understanding of ocular dominance column formation. A reduced mathematical model is constructed based on the thermodynamics in the Ising spin variables representing the afferent synaptic connection distribution. The results of Monte Carlo simulations on the segregation of ipsilateral and contralateral synaptic terminals in the input layer of the primary visual cortex suggest the existence of phase transition phenomena. Three types of ocular dominance column patterns — stripe, blob, and uniform — are visualized according to the values of the correlation strength and the degree of imbalance in activity between the left and right retinas. The theory presented here successfully explains how ocular dominance columns are developed.     

Electrical processes involved in the encoding of nerve impulses

A mechanism for impulse encoding is advanced for those neurones whose impulse trigger zone membrane is more excitable than the general axonal membrane. Electrical communication between an electrotonically small patch of highly excitable membrane and neighboring membrane places the control of membrane potential — in varying degree — to the larger membrane area throughout the interspike intervals. That control is relinquished to the trigger membrane near the time of action potential initiation in a natural fashion. Model calculations demonstrate that this mechanism can lead to a dramatic lowering of the minimum stable firing frequency of tonic neurons, and, additionally influence the shape of the stimulus —versus — impulse frequency curve. The results are compared with the behavior of the slowly adapting stretch receptor neuron of the crayfish.Research supported by NSF grant BNS 77-22532 and Public Health Service Grant EY 00293. Computer facilities were made available by the Air Force Office of Scientific Research (AFOSR-1221) and by the University of Minnesota Computer Center     

Excitatory synapses of blue crab gastric mill muscles

Summary Physiological and ultrastructural studies were made of neuromuscular synapses in stomach muscles, especially two gastric mill muscles of the blue crab innervated by neurons of the stomatogastric ganglion. These muscles depolarized and contracted with application of glutamate, but not acetylcholine, whereas the dorsal dilator muscles of the pyloric region depolarized and contracted in acetylcholine, but not in glutamate. Large excitatory postsynaptic potentials (EPSP's) of 5–20 mV were recorded in the gastric mill muscles. At low frequencies of activation, individual synapses released on average about 2 quanta of transmitter for each nerve impulse. Facilitation of EPSP's after a single nerve impulse could be detected for at least 10 s. Synapses were found on enlarged terminals of the motor axon; their contact areas ranged from 0.2 m² up to 3 m². Both electron-lucent, round synaptic vesicles and dense-cored vesicles occurred near these synapses. A possible correlation between contact area of a synapse and output of transmitter, is discussed.Supported by grants from the National Research Council of Canada and the Muscular Dystrophy Association of Canada to H.L. Atwood and C.K. Govind. We thank Kazuko Hay, Eva Yap-Chung and Irene Kwan for technical assistance with electron microscopy and reconstruction of nerve terminals from micrographs     

QUANTITATIVE ASPECTS OF TRANSMITTER RELEASE

The opener-stretcher motor neuron in crayfish makes 50 endings upon each of 1200 muscle fibers. We have calculated the quantal content of junctional potentials produced by individual terminals and by the whole cell at various physiological frequencies. The results show that when the motor neuron is active at 20 impulses/second, it releases 50 quanta/impulse per muscle fiber, or a total of 4.5 x 10⁹ quanta/hr. These figures are similar to those for vertebrate muscles per fiber, but larger for the entire neuron because the opener motor unit is so large. On the basis that the quanta correspond to synaptic vesicles each containing 10³–10⁴ molecules of transmitter, the release rate must be around 10^-11 mole/hr. This value is within an order of magnitude of the release figures obtained for mammalian neurons by collecting transmitter in perfusates, but it is far lower than the value reported for a crustacean inhibitory neuron. If the membrane materials surrounding each vesicle were lost in the release process, the replacement synthesis would involve 24 mm² of membrane/hr. We conclude that the metabolic load in terms of transmitter synthesis is probably sustainable, but that the release mechanism must operate in such a way that vesicle membrane materials are neither lost nor incorporated into the terminal membrane.     

Statistical Estimation of Distribution of Quantal Release of Transmitter in Neuromuscular Synapse of the Lobster Homarus americanus

A new approach to estimation of quantal release distribution of transmitter under conditions of high synaptic activity is presented. Postsynaptic responses of neuromuscular excitatory synapse in muscle-opener of nipper of the lobster, which are obtained by focal extracellular recording, are used as original data set. Based on two data groups (value of evoked and spontaneous postsynaptic responses), the linear regression model is constructed. Parameters of this model describe completely the quantal release distribution. To evaluate the parameters, biased modifications of the least squares method—the penalized least squares method and the principal components method—were applied. As a result, it was possible to achieve estimations of the quantal release distribution with sufficiently low standard errors. Modeling studies have shown that the gain of accuracy of the estimation due to a decrease of the standard error exceeds considerably losses caused by its bias.     

Discharges of tectal neurons evoked by electrical stimulation of single retinal ganglion cells in frogs

"EEG quanta" — extracellular monosynaptic PSPs evoked by action potentials of a single axon — were recorded in the frog thalamus at a depth of 200–700 µ during electrical stimulation of the retina by single, double, or triple pulses of current of threshold strength. With an interval of 5–25 msec between consecutive stimuli, a negative spike, increasing as the microelectrode was inserted deeper, was observed at the peak of the testing EEG quanta, which were enlarged 1.5–2.5 times. It is postulated that this spike is the synchronous discharge of neuron bodies (population spike), evoked by an action potential of a single retinotectal axon. This axon branches in layers F or G. Discharges appear in neurons of layers 6–8. The possibility that the volley discharge of one class 3–5 detector excites tectal ganglion cells, whose axons mainly form the tecto-bulbospinal tract, is discussed.Research Institute of Cardiology and Laboratories of Electroencephalography and Neurocybernetics, Medical Institute, Kaunas, Lithuania. Translated from Neirofiziologiya, Vol. 16, No. 6, pp. 829–835, November–December, 1984.     

Trigger regime of the functioning of a synaptic channel

It has been shown using the linear model describing the dynamics of biochemical reactions occurring in a synapse that, as periodic step-like impulses pass through a synaptic channel, it operates in the triggering regime. The transmission of an impulse through the channel is associated with the change of the stationary point of the corresponding dynamic system compared with the unexcited state of the synapse. As a result, the system periodically evolves between two stable stationary states.     

Effect of filaments within the synaptic cleft on the response of excitatory synapses simulated by computer experiments

Mathematical models of the excitatory synapse are furnishing valuable information about the synaptic response. Based on Brownian-diffusion of glutamate molecules, a synapse model was utilized to investigate the synaptic response on a femto-second time scale by the use of a parallel computer. In particular, the presence of fibrils crossing the synaptic cleft was simulated, which could have a role in shaping the brain activity. To this aim the model of synapse was modified by considering trans-synaptic filaments with diameters ranging from 7 nm to 3 nm, disposed on a grid with spacing of 14 nm or 8 nm. The simulation demonstrated that the presence of filaments induced an increase in the synaptic response, most likely linked to an increment in the probability of encounter between glutamate molecules and receptors. The increase was small - from 5 to 20%, but metabolic and functional considerations provide substantive hints about the importance of these small changes for brain activity. Moreover, it was shown that the presence of filaments made more stable the response of the synapse to random variations of pre-synaptic elements. Originated by these computational results, some inferences about the biological bases of mind diseases such as autism, mental retardation and schizophrenia, are reported in the Discussion.