Key principles

Schematic of a human brain, with some key landmarks/structures

Background

The brain is a fascinating device, that remains to be understood. Made of approximatly neurons, each of them connected to about 10000 targets, it is thus a complex and unique network with more than connections interacting with each others in an efficient and robust manner. We all have the same organization at a macroscopic scale, the same generic cotical areas: sensory inputs coming from our senses are wired in a reproducible manner at a large scale, but the fine details of those connections, that are making each of us unique, are still unknown.

Understanding how the primary sensory areas of the neocortex are structured in order to process sensory inputs is a crucial step in analysing the mechanisms underlying the functional role, from an algorithmic point of view, of cerebral activity. This understanding of the sensory dynamics, at a large scale level, implies using simplified models of neurons, such as the "integrate-and-fire models", and a particular framework, the "balanced" random network, which allows the recreation of dynamical regimes of conductances close to those observed in vivo, in which neurons spike at low rates and with an irregular discharge.

The Neuron 

Neurons are, with glia cells, the fundamental processing units of the brain. Because of their fast time constants, they are considered as reponsible for most of the information processing, while the latter are thought to be involved in much slower mechanisms such as plasticity or metabolism. Therefore, as a (pretty strong) first assumption, they will be ignored in the following.

Neurons are basically acting as spatio-temporal integrators, exchanging binary information through action potentials: they are receiving inputs from pre-synaptic cells, and when "enough" are received, either temporally or spatially, summed with complex non-linear interaction depending on their morphologies, they emit an action potential and send it to their targets. To picture it, imagine a bath tube, with a hole at the bottom, and thus leeking water. Incoming action potentials are like buckets of water that will be poured in time into the bath tube. If enough are incoming during a short period of time (and thus compensating for the leak), water will overflow. The neuron is emitting an action potential, and its voltage (equivalent here to the water level) will be reset to a default value. This is the principe of the "integrate and fire neurons". Of course, this model is a crude simplication of all the biological processes that are taking place in a real neurons. It is discarding all the non-linearities and the complex operations that may be achieved in the dendrites. However, because of its tractability, either from a computational or an analytical point of view, it is widely used in computational neuroscience.

First hand drawings of cortical neurons, drawn by Ramon y Cajal

The integrate-and-fire model of Lapicque. (A) The equivalent circuit with membrane capacitance and membrane resistance . is the membrane potential, is the resting membrane potential, and is an injected current. (B) The voltage trajectory of the model. When reaches a threshold value, an action potential is generated and is reset to a subthreshold value. (C) An integrate-and-fire model neuron driven by a timevarying current. The upper trace is the membrane potential and the bottom trace is the input current.

The Integrate-and-fire model

From a more mathematical point of view, inputs to the neurons are described as ionic currents flowing through the cell membrane when neurotransmitters are released. Their sum is seen as a physical time-dependent current and the membrane is described as an circuit, charged by (see Figure taken from [Abbott1999]). When the membrane potential reaches a threshold value , a spike is emitted and the membrane potential is reset. In its basic form, the equation of the integrate and fire model is:

\begin{equation}
\tau_{\mathrm{m}} \frac{dV_{\mathrm{m}}(t)}{dt} = -V_{\mathrm{m}}(t) + RI(t)
\end{equation}

where is the resistance of the membrane, with .

To refine and be more precise, the neuronal input approximated as a fluctuating current but synaptic drives are better modelled by fluctuating conductances: the amplitudes of the post synaptic potentials (PSP) evoked by neurotransmitter release from pre-synaptic neuron depend on the post-synaptic depolarization level. A lot of study focuses now on this integrate-and-fire model with conductance-based synapses [Destexhe2001, Tiesinga2000, Cessac2008, Vogels2005]. The equation of the membrane potential dynamic is then:

\begin{equation}
\tau_{\mathrm{m}} \frac{dV_{\mathrm{m}}(t)}{dt} = (V_{\mathrm{rest}}-V_{\mathrm{m}}(t)) + g_{\mathrm{exc}}(t)(E_{\mathrm{exc}}-V_{\mathrm{m}}(t)) + g_{\mathrm{inh}}(t)(E_{\mathrm{inh}}-V_{\mathrm{m}}(t))
\end{equation}

When reaches the spiking threshold , a spike is generated and the membrane potential is held at the resting potential for a refractory period of duration . Synaptic connections are modelled as conductance changes: when a spike is emitted followed by exponential decay with time constants and for excitatory and inhibitory post-synaptic potentials, respectively. The shape of the PSP may not be exponential. Other shapes for the PSP can be used, such as alpha synapses , or double shaped exponentials synapses . and are the reversal potentials for excitation and inhibition.

The chemical synapse

The synapse is a key element where the axon of a pre-synaptic neuron connects with the dendritic arbour of a post-synaptic neuron . It transmit the electrical influx emitted by neuron to .  Synapses are crucial in shaping a network's structure, and their ability to modify their efficacy according to the activity of the pre and the post-synaptic neuron is at the origin of synaptic plasticity and memory retention in neuronal networks.

Synapses can be either chemical or electrical, but again, for a more exhaustive description,the latter here will be discarded. To focus only on the chemical synapses, the pre-synaptic neuron releases a neurotransmitter into the synaptic cleft which then binds to receptors located on the surface of the post-synaptic neuron , embedded in the plasma membrane. These neurotransmitters are stored in vesicles, regenerated continuously, but a too strong stimulation of the synapse may lead to a temporary lack of neurotransmitter, or to a saturation of the post-synaptic receptors on . This short-term plasticity phenomenon is called synaptic adaptation.

The type of neurotransmitter which is received to the post-synaptic neuron influences its activity. The synaptic current is cancelled for a given inversion potential: if this inversion potential is below (the voltage threshold for triggering an action potential), the net synaptic effect inhibits the neuron, and if it is below, it excits the cell. Classical neurotransmitter such as glutamate leads to a depolarization (i.e. an increase of the membrane potential), and the synapse is said to be excitatory. In contrast, gamma-aminobutyric acid (GABA) leads to an hyper-polarization (a decrease of the membrane potential), and the synapse is said to be inhibitory. In general, a given neuron produces only one type of neurotransmitter, being either only excitatory or only inhibitory. This principle is known as the Dale's principle, and is a common assumption made in the models of neuronal networks.

 

 

 

Top: schematic illustration of a synaptic contact between two neurons. The axon of pre-synaptic neuron establishes a synapse with a dendrite of post-synaptic neuron $B$. Bottom: detail of the synaptic cleft. Neurotransmitters stored in vesicles are liberated when the pre-synaptic membrane is depolarized, and then docked onto receptors of

References

[Abbott1999] [doi] L. F. Abbott, "Lapicque’s introduction of the integrate-and-fire model neuron (1907)," Brain research bulletin, vol. 50, iss. 5-6, pp. 303-304, 1999.
[Bibtex]
@article{Abbott1999,
author = {Abbott, L.F},
doi = {10.1016/S0361-9230(99)00161-6},
file = {:home/pierre/Mendeley/Abbott - 1999.pdf:pdf},
issn = {03619230},
journal = {Brain Research Bulletin},
month = nov,
number = {5-6},
pages = {303--304},
title = {{Lapicque’s introduction of the integrate-and-fire model neuron (1907)}},
url = {http://linkinghub.elsevier.com/retrieve/pii/S0361923099001616},
volume = {50},
year = {1999}
}
[Destexhe2001] a Destexhe, M. Rudolph, J. M. Fellous, and T. J. Sejnowski, "Fluctuating synaptic conductances recreate in vivo-like activity in neocortical neurons.," Neuroscience, vol. 107, iss. 1, pp. 13-24, 2001.
[Bibtex]
@article{Destexhe2001,
abstract = {To investigate the basis of the fluctuating activity present in neocortical neurons in vivo, we have combined computational models with whole-cell recordings using the dynamic-clamp technique. A simplified 'point-conductance' model was used to represent the currents generated by thousands of stochastically releasing synapses. Synaptic activity was represented by two independent fast glutamatergic and GABAergic conductances described by stochastic random-walk processes. An advantage of this approach is that all the model parameters can be determined from voltage-clamp experiments. We show that the point-conductance model captures the amplitude and spectral characteristics of the synaptic conductances during background activity. To determine if it can recreate in vivo-like activity, we injected this point-conductance model into a single-compartment model, or in rat prefrontal cortical neurons in vitro using dynamic clamp. This procedure successfully recreated several properties of neurons intracellularly recorded in vivo, such as a depolarized membrane potential, the presence of high-amplitude membrane potential fluctuations, a low-input resistance and irregular spontaneous firing activity. In addition, the point-conductance model could simulate the enhancement of responsiveness due to background activity. We conclude that many of the characteristics of cortical neurons in vivo can be explained by fast glutamatergic and GABAergic conductances varying stochastically.},
author = {Destexhe, a and Rudolph, M and Fellous, J M and Sejnowski, T J},
file = {:home/pierre/Mendeley/Destexhe et al. - 2001.pdf:pdf},
issn = {0306-4522},
journal = {Neuroscience},
keywords = {Action Potentials,Action Potentials: drug effects,Action Potentials: physiology,Animals,Cats,Cell Compartmentation,Cell Compartmentation: physiology,Dendrites,Dendrites: physiology,Glutamic Acid,Glutamic Acid: metabolism,Ion Channels,Ion Channels: drug effects,Ion Channels: physiology,Models, Neurological,Neocortex,Neocortex: cytology,Neocortex: drug effects,Neocortex: physiology,Nerve Net,Nerve Net: drug effects,Nerve Net: physiology,Neural Inhibition,Neural Inhibition: drug effects,Neural Inhibition: physiology,Organ Culture Techniques,Patch-Clamp Techniques,Pyramidal Cells,Pyramidal Cells: cytology,Pyramidal Cells: drug effects,Pyramidal Cells: physiology,Rats,Rats, Sprague-Dawley,Receptors, AMPA,Receptors, AMPA: drug effects,Receptors, AMPA: physiology,Stochastic Processes,Synapses,Synapses: drug effects,Synapses: physiology,Synaptic Transmission,Synaptic Transmission: drug effects,Synaptic Transmission: physiology,Tetrodotoxin,Tetrodotoxin: pharmacology,gamma-Aminobutyric Acid,gamma-Aminobutyric Acid: metabolism},
month = jan,
number = {1},
pages = {13--24},
pmid = {11744242},
title = {{Fluctuating synaptic conductances recreate in vivo-like activity in neocortical neurons.}},
url = {http://www.pubmedcentral.nih.gov/articlerender.fcgi?artid=3320220\&tool=pmcentrez\&rendertype=abstract},
volume = {107},
year = {2001}
}
[Tiesinga2000] P. H. Tiesinga, J. V. José, and T. J. Sejnowski, "Comparison of current-driven and conductance-driven neocortical model neurons with Hodgkin-Huxley voltage-gated channels.," Physical review. e, statistical physics, plasmas, fluids, and related interdisciplinary topics, vol. 62, iss. 6 Pt B, pp. 8413-9, 2000.
[Bibtex]
@article{Tiesinga2000,
abstract = {Intrinsic noise and random synaptic inputs generate a fluctuating current across neuron membranes. We determine the statistics of the output spike train of a biophysical model neuron as a function of the mean and variance of the fluctuating current, when the current is white noise, or when it derives from Poisson trains of excitatory and inhibitory postsynaptic conductances. In the first case, the firing rate increases with increasing variance of the current, whereas in the latter case it decreases. In contrast, the firing rate is independent of variance (for constant mean) in the commonly used random walk, and perfect integrate-and-fire models for spike generation. The model neuron can be in the current-dominated state, representative of neurons in the in vitro slice preparation, or in the fluctuation-dominated state, representative of in vivo neurons. We discuss the functional relevance of these states to cortical information processing.},
author = {Tiesinga, P H and Jos\'{e}, J V and Sejnowski, T J},
file = {:home/pierre/Mendeley/Tiesinga, Jos\'{e}, Sejnowski - 2000.pdf:pdf},
issn = {1063-651X},
journal = {Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics},
keywords = {Acoustic Stimulation,Animals,Biophysical Phenomena,Biophysics,Electric Conductivity,Electric Stimulation,Evoked Potentials,Ion Channels,Ion Channels: physiology,Models, Neurological,Neocortex,Neocortex: physiology,Neurons,Neurons: physiology},
month = dec,
number = {6 Pt B},
pages = {8413--9},
pmid = {11138142},
title = {{Comparison of current-driven and conductance-driven neocortical model neurons with Hodgkin-Huxley voltage-gated channels.}},
url = {http://www.ncbi.nlm.nih.gov/pubmed/11138142},
volume = {62},
year = {2000}
}
[Cessac2008] [doi] B. Cessac and T. Viéville, "On dynamics of integrate-and-fire neural networks with conductance based synapses.," Frontiers in computational neuroscience, vol. 2, p. 2, 2008.
[Bibtex]
@article{Cessac2008,
abstract = {We present a mathematical analysis of networks with integrate-and-fire (IF) neurons with conductance based synapses. Taking into account the realistic fact that the spike time is only known within some finite precision, we propose a model where spikes are effective at times multiple of a characteristic time scale delta, where delta can be arbitrary small (in particular, well beyond the numerical precision). We make a complete mathematical characterization of the model-dynamics and obtain the following results. The asymptotic dynamics is composed by finitely many stable periodic orbits, whose number and period can be arbitrary large and can diverge in a region of the synaptic weights space, traditionally called the "edge of chaos", a notion mathematically well defined in the present paper. Furthermore, except at the edge of chaos, there is a one-to-one correspondence between the membrane potential trajectories and the raster plot. This shows that the neural code is entirely "in the spikes" in this case. As a key tool, we introduce an order parameter, easy to compute numerically, and closely related to a natural notion of entropy, providing a relevant characterization of the computational capabilities of the network. This allows us to compare the computational capabilities of leaky and IF models and conductance based models. The present study considers networks with constant input, and without time-dependent plasticity, but the framework has been designed for both extensions.},
author = {Cessac, Bruno and Vi\'{e}ville, Thierry},
doi = {10.3389/neuro.10.002.2008},
file = {:home/pierre/Mendeley/Cessac, Vi\'{e}ville - 2008.pdf:pdf},
issn = {1662-5188},
journal = {Frontiers in computational neuroscience},
keywords = {generalized integrate and fi,neural code,neural networks dynamics,re models,spiking network},
month = jan,
pages = {2},
pmid = {18946532},
title = {{On dynamics of integrate-and-fire neural networks with conductance based synapses.}},
url = {http://www.pubmedcentral.nih.gov/articlerender.fcgi?artid=2525942\&tool=pmcentrez\&rendertype=abstract},
volume = {2},
year = {2008}
}
[Vogels2005] [doi] T. P. Vogels, K. Rajan, and L. F. Abbott, "Neural network dynamics.," Annu rev neurosci, vol. 28, pp. 357-376, 2005.
[Bibtex]
@article{Vogels2005,
abstract = {Neural network modeling is often concerned with stimulus-driven responses, but most of the activity in the brain is internally generated. Here, we review network models of internally generated activity, focusing on three types of network dynamics: (a) sustained responses to transient stimuli, which provide a model of working memory; (b) oscillatory network activity; and (c) chaotic activity, which models complex patterns of background spiking in cortical and other circuits. We also review propagation of stimulus-driven activity through spontaneously active networks. Exploring these aspects of neural network dynamics is critical for understanding how neural circuits produce cognitive function.},
author = {Vogels, Tim P and Rajan, Kanaka and Abbott, L F},
doi = {10.1146/annurev.neuro.28.061604.135637},
file = {:home/pierre/Mendeley/Vogels, Rajan, Abbott - 2005.pdf:pdf},
issn = {0147-006X},
journal = {Annu Rev Neurosci},
keywords = {Action Potentials; Animals; Humans; Nerve Net; Neu,Computer-Assisted; Time Factors,Extramural; Research Support,N.I.H.,Non-P.H.S.; Research Support,Non-U.S. Gov't; Research Support,P.H.S.; Signal Processing,U.S. Gov't},
pages = {357--376},
pmid = {16022600},
shorttitle = {Annu Rev Neurosci},
title = {{Neural network dynamics.}},
url = {http://dx.doi.org/10.1146/annurev.neuro.28.061604.135637},
volume = {28},
year = {2005}
}

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