Single-compartment Neurons Research Articles

Bringing rest into consideration: analyzing a database of computational models for multistability of oscillatory and stationary regimes

Biophysically realistic neuronal models are defined in high-dimensional parameter spaces. Brute-force database approaches have proven themselves fruitful in characterizing neuronal models [1,3,5]. In a brute-force database approach, the corresponding activities and parameters of the simulated neuronal models are stored in a database, which is further efficiently searched for insightful information about the models. We present a novel technique which combines a brute-force approach with bifurcation continuation methods from the theory of dynamical systems. We considered a database of single compartment neuron model that can display, silence, tonic spiking and bursting activity [3]. A one-parameter bifurcation analysis allowed us to determine stable and unstable stationary states for each case from the database. Using this novel database technique, we systematically tested models for the coexistence of stable stationary states and bursting activity. Specifically we used this technique to extend the database describing the model of the isolated leech heart interneuron to include stationary states [3]. We took into consideration only those cases that were shown to produce functional half-center oscillators (HCOs) [2-4]. By using the leak conductance (gleak) as the controlling parameter for bifurcation analysis, we showed that the model could exhibit two stable depolarized stationary states and two hyperpolarized stationary states in addition to various spiking and bursting regimes. For different initial conditions as many as three different attracting regimes could be exhibited by a case of the model. We found cases demonstrating either bursting activity, hyperpolarized stationary state or depolarized stationary state. We investigated the 2,387 cases that produced regular bursting activity, and found that 421 (18%) were multistable. We found that among the latter there were 361 cases which exhibited bistability of bursting and a hyperpolarized stationary state and 47 cases which showed bistability of bursting and a depolarized stationary state. Out of those 361 cases, one case showed bistability of bursting and a second, slightly less hyperpolarized stationary state. Twelve cases demonstrated tristability of coexisting bursting and two stable stationary states, hyperpolarized and depolarized. Our method also allowed us to determine that 2,128 out of 2,387 (89%) cases are multistable for some range of gleak values. Among these cases, the majority, 2,078 (98%), showed at least one range of gleak values with two attracting regimes, bursting and a hyperpolarized stationary state. We also explored the activity of HCOs built of multistable regular bursting model neurons (421 described above) in response to perturbations of the state variables. For a given set of initial conditions, 353 (84%) cases showed functional activity. We further explored the activity of these 353 cases. After 100 seconds of functional activity the state of one of the neurons was reset to be close to a stable stationary state, and the synaptic conductances determining the mutual inhibition of the HCO were set to zero for 5 seconds. This interruption of the normal HCO dynamics produced a perturbation of the state variables. In 96% of these cases, the HCOs exhibited functional bursting, proving itself robust against this perturbation too. Multistability of oscillatory regimes and stationary states poses a threat to functionality of central pattern generators which are required to produce a persistent pattern. The half-center oscillator configuration could serve as network-based protective mechanism against malfunctional multistability. On the other hand multistability of the dynamics of neurons could be functional providing a substrate for memory units, toggle switches, and elements of multifunctional central pattern generators. BM is supported by the Brazilian agency FAPESP. GC and WB are supported by NSF grant PHY-0750456. AD-M and RLC are supported by NIH grant NS024072

Phase-dependent stimulation effects on bursting activity in a neural network cortical simulation

A neural network simulation with realistic cortical architecture has been used to study synchronized bursting as a seizure representation. This model has the property that bursting epochs arise and cease spontaneously, and bursting epochs can be induced by external stimulation. We have used this simulation to study the time-frequency properties of the evolving bursting activity, as well as effects due to network stimulation. The model represents a cortical region of 1.6 mm x 1.6mm, and includes seven neuron classes organized by cortical layer, inhibitory or excitatory properties, and electrophysiological characteristics. There are a total of 65,536 modeled single compartment neurons that operate according to a version of Hodgkin-Huxley dynamics. The intercellular wiring is based on histological studies and our previous modeling efforts. The bursting phase is characterized by a flat frequency spectrum. Stimulation pulses are applied to this modeled network, with an electric field provided by a 1mm radius circular electrode represented mathematically in the simulation. A phase dependence to the post-stimulation quiescence is demonstrated, with local relative maxima in efficacy occurring before or during the network depolarization phase in the underlying activity. Brief periods of network insensitivity to stimulation are also demonstrated. The phase dependence was irregular and did not reach statistical significance when averaged over the full 2.5s of simulated bursting investigated. This result provides comparison with previous in vivo studies which have also demonstrated increased efficacy of stimulation when pulses are applied at the peak of the local field potential during cortical after discharges. The network bursting is synchronous when comparing the different neuron classes represented up to an uncertainty of 10 ms. Studies performed with an excitatory chandelier cell component demonstrated increased synchronous bursting in the model, as predicted from experimental work. This large-scale multi-neuron neural network simulation reproduces many aspects of evolving cortical bursting behavior as well as the timing-dependent effects of electrical stimulation on that bursting.

The effects of cholinergic neuromodulation on neuronal phase-response curves of modeled cortical neurons.

The response of an oscillator to perturbations is described by its phase-response curve (PRC), which is related to the type of bifurcation leading from rest to tonic spiking. In a recent experimental study, we have shown that the type of PRC in cortical pyramidal neurons can be switched by cholinergic neuromodulation from type II (biphasic) to type I (monophasic). We explored how intrinsic mechanisms affected by acetylcholine influence the PRC using three different types of neuronal models: a theta neuron, single-compartment neurons and a multi-compartment neuron. In all of these models a decrease in the amount of a spike-frequency adaptation current was a necessary and sufficient condition for the shape of the PRC to change from biphasic (type II) to purely positive (type I).

Brian: a simulator for spiking neural networks in Python

is a new simulator for spiking neural networks, written in Python (http://brian.di.ens.fr). It is an intuitive and highly flexible tool for rapidly developing new models, especially networks of single-compartment neurons. In addition to using standard types of neuron models, users can define models by writing arbitrary differential equations in ordinary mathematical notation. Python scientific libraries can also be used for defining models and analysing data. Vectorisation techniques allow efficient simulations despite the overheads of an interpreted language. Brian will be especially valuable for working on non-standard neuron models not easily covered by existing software, and as an alternative to using Matlab or C for simulations. With its easy and intuitive syntax, Brian is also very well suited for teaching computational neuroscience.

Two Computational Regimes of a Single-Compartment Neuron Separated by a Planar Boundary in Conductance Space

Recent in vitro data show that neurons respond to input variance with varying sensitivities. Here we demonstrate that Hodgkin-Huxley (HH) neurons can operate in two computational regimes: one that is more sensitive to input variance (differentiating) and one that is less sensitive (integrating). A boundary plane in the 3D conductance space separates these two regimes. For a reduced HH model, this plane can be derived analytically from the V nullcline, thus suggesting a means of relating biophysical parameters to neural computation by analyzing the neuron's dynamical system.

Two Computational Regimes of a Single-Compartment Neuron Separated by a Planar Boundary in Conductance Space

Recent in vitro data show that neurons respond to input variance with varying sensitivities. Here we demonstrate that Hodgkin-Huxley (HH) neurons can operate in two computational regimes: one that is more sensitive to input variance (differentiating) and one that is less sensitive (integrating). A boundary plane in the 3D conductance space separates these two regimes. For a reduced HH model, this plane can be derived analytically from the V nullcline, thus suggesting a means of relating biophysical parameters to neural computation by analyzing the neuron's dynamical system.

Brian: a simulator for spiking neural networks in Python

“Brian” is a new simulator for spiking neural networks, written in Python (http://brian. di.ens.fr). It is an intuitive and highly flexible tool for rapidly developing new models, especially networks of single-compartment neurons. In addition to using standard types of neuron models, users can define models by writing arbitrary differential equations in ordinary mathematical notation. Python scientific libraries can also be used for defining models and analysing data. Vectorisation techniques allow efficient simulations despite the overheads of an interpreted language. Brian will be especially valuable for working on non-standard neuron models not easily covered by existing software, and as an alternative to using Matlab or C for simulations. With its easy and intuitive syntax, Brian is also very well suited for teaching computational neuroscience.

A parameter-space search algorithm tested on a Hodgkin–Huxley model

We demonstrate a parameter-space search algorithm using a computational model of a single-compartment neuron with conductance-based Hodgkin-Huxley dynamics. To classify bursting (the desired behavior), we use a simple cost function whose inputs are derived from the frequency content of the neural output. Our method involves the repeated use of a stochastic gradient descent-type algorithm to locate parameter values that allow the neural model to produce bursting within a specified tolerance. We demonstrate good results, including those showing that the utility of our algorithm improves as the pre-defined allowable parameter ranges increase and that the initial approach to our method is computationally efficient.

Parameter estimation in single-compartment neuron models using a synchronization-based method

We describe a method for estimating parameter values in single-compartment neuron models from a recorded voltage trace. The technique involves synchronization between the model and the experimental data and minimizing a cost function that depends on the goodness of this synchrony with a minimum for optimal parameter values. We test our method using a previously developed single-compartment interneuron model that includes noise to mimic experimental data. Direct computation of the cost function over ranges of parameter values suggests that it changes smoothly, allowing for downhill optimization routines that utilize the gradient. We implement one such optimization routine with several kinetic parameters set at incorrect values. The optimization is fast and consistently returns good values. We discuss the considerations necessary when applying this approach to noisy data. In conclusion, we believe that this approach will be a useful tool in developing neuronal models.

A single compartment neuron model with activity-dependent conductances during NMDA induced activity

In most neuron models the values of maximal conductances of membrane ionic currents are fixed. In our paper we investigate spiking activity of the neuron model activated tonically by NMDA synapse, when the membrane ionic currents are dynamically dependent on calcium concentration, as in a model by Abbott and coauthors (1993). A spiking neuron model (in Matlab/Simulink environment) is based on the properties of lamprey spinal neurons. The basic neuron is a one-compartment model with voltage-gated Na+, K+, Ca2+, KCa+ channels. The Na+ and K+ currents are described with the dynamic equations of Hodgkin-Huxley model (Hodgkin and Huxley 1952). The Ca2+ and KCa+ channels are modeled using description of calcium dynamic introduced by Ekeberg and coauthors (1991). The model was tonically activated by NMDA synapse described by a kinetic model of synaptic transmission. We analyzed the activity of this model and showed that when only one of conductances is calcium-dependent, the cell is not able to react to and recover from external perturbations.

Minimal Models of Bursting Neurons: How Multiple Currents, Conductances, and Timescales Affect Bifurcation Diagrams

After reviewing the Hodgkin--Huxley ionic current formulation, we introduce a three-variable generic model of a single-compartment neuron comprising a two-dimensional fast subsystem and a very slow...

Reduction of intracellular calcium removal rate can explain changes in seizure dynamics: studies in neuronal network models

Complex partial seizures originating from mesial temporal structures are characterized by relatively short durations of organized rhythmic activity (ORA) of 5–8 Hz, typically lasting less than 60 s. Previous investigations into seizure dynamics have revealed that this ORA undergoes a monotonic decline prior to seizure evolution into intermittent bursting and subsequent seizure termination. Large neural network models of simplified single-compartment neurons were employed to address the hypothesis that changes in the free intracellular calcium ([Ca 2+] i) removal rate during network bursting can result in the alterations of rhythmic seizure activity similar to that observed in recordings from humans. Both exponential and linear models of decreasing calcium removal rates resulted in changes in the predominant frequency of network bursting very similar in frequency and time course to those seen in human intracranial recordings. This supports the concept that changes in [Ca 2+] i removal can explain this important network behavior, while not excluding alternative hypotheses. Identifying potential mechanisms underlying the dynamic changes seen in epileptogenic activity in large neural networks can provide important insights into seizure evolution and termination. Model neural network ensembles are attractive systems to address these questions that are difficult to investigate in biological preparations.

A bursting mechanism of chattering neurons based on Ca 2+-dependent cationic currents

We propose an ionic conductance model that simulates fast rhythmic bursts in a single compartment neuron. In this modeling, the essential point is that Ca 2+-dependent cationic current, whose reversal potential is approximately −45 mV , plays a key role for generating the depolarizing afterpotential (DAP) and the doublet/triplet firing. For the calcium dynamics, the kinetics of the extrusion and the chelation of intracellular free Ca 2+ with Ca 2+-pump and buffering proteins is taken into account. The resulting model quite accurately predicts the experimentally observed natural pattern of the chattering behavior.

Reinforcement learning by Hebbian synapses with adaptive thresholds

A central problem in learning theory is how the vertebrate brain processes reinforcing stimuli in order to master complex sensorimotor tasks. This problem belongs to the domain of supervised learning, in which errors in the response of a neural network serve as the basis for modification of synaptic connectivity in the network and thereby train it on a computational task. The model presented here shows how a reinforcing feedback can modify synapses in a neuronal network according to the principles of Hebbian learning. The reinforcing feedback steers synapses towards long-term potentiation or depression by critically influencing the rise in postsynaptic calcium, in accordance with findings on synaptic plasticity in mammalian brain. An important feature of the model is the dependence of modification thresholds on the previous history of reinforcing feedback processed by the network. The learning algorithm trained networks successfully on a task in which a population vector in the motor output was required to match a sensory stimulus vector presented shortly before. In another task, networks were trained to compute coordinate transformations by combining different visual inputs. The model continued to behave well when simplified units were replaced by single-compartment neurons equipped with several conductances and operating in continuous time. This novel form of reinforcement learning incorporates essential properties of Hebbian synaptic plasticity and thereby shows that supervised learning can be accomplished by a learning rule similar to those used in physiologically plausible models of unsupervised learning. The model can be crudely correlated to the anatomy and electrophysiology of the amygdala, prefrontal and cingulate cortex and has predictive implications for further experiments on synaptic plasticity and learning processes mediated by these areas.

Comparison of three different methods for target detection and tracking

We introduce a biologically based method for motion detection and track initiation. The model consists of five hexagonally packed layers of single compartment neurons. The model mimics some of the processing in the retina and has localized interactions between cells in each layer. To evaluate the gain from using a biological model versus a more conventional approach we have compared the performance of three methods: simple intensity thresholding, the biologically inspired model, and an algorithm involving a truncated sequential probability ratio test. These methods were tested with a large number of real and simulated astronomical data and with a standard set of 50 images using targets at various SNRs. We discuss the importance of various aspects of the biological model for the overall performance on a target detection task.