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It also covers glycoproteins from four more species slime mold, snails, fish, batracians. The content of the volume is very comprehensive in that most contributors are focussed on discussing, in depth, the wealth of most recent advances in their field, referring to previous reviews of older work for background information. The volume is an important information source for all glycobiologist researchers senior investigators, post-doctoral fellows and graduate students , and as a good, comprehensive, reference text for scientists working in the life sciences.

Glycoproteins I J Montreuil, J F G Vliegenthart, H Schachter Part I covers modern advances in the determination ofglycoprotein structure and in the biosynthesis of mammalian,bacterial, yeast, plant and insect glycoproteins. Glycoproteins and Disease J Montreuil The elucidation during the latter half of the 20th century of the mechanisms by which information flows from nucleic acids to proteins has completely changed the face of biological research.

Bloggat om Glycoproteins II. The concept of network as a mathematical description of a set of states, or events, linked according to a certain topology has been developed recently and has led to a novel approach of real world. This approach is no doubt important in the field of biology. In fact biological systems can be considered networks. Thus, for instance, an enzyme-catalysed reaction is a network that links, according to a certain topology, the various states of the protein and of its complexes with the substrates and products of the chemical reaction.

These equations take the form of a linear equation, complemented by a sigmoidal coupling function. These equations have recently been generalized to take fluctuations and higher-order moments into account Buice and Cowan, ; Buice et al. Already in the original paper by Wilson and Cowan arguments have been given that guarantee that fast dynamics on the time-scale of the refractory period does not affect the population dynamics.

Different rate equations have been derived by resorting to ensemble averaging rather than temporal coarse-graining for the case of spiking neurons with refractoriness Deger et al. Indeed, the refractory period was found to have a strong effect on the population dynamics. A very successful, and by now classical theory for studying rate dynamics in homogeneous networks of integrate-and-fire neurons is based on a Fokker-Planck approach Abbott and van Vreeswijk, ; Fusi and Mattia, ; Brunel, ; Knight et al. Some results about correlations arising from network interactions have recently been obtained, though Ostojic et al.

We will not follow this approach here, since the use of second-order partial differential equations makes the study of network dynamics difficult. Ensemble averaging is the preferred approach for studying time-evolution of activity in neural networks. Ensemble averaging can be used to study neural dynamics both on the single cell and on the population level Kriener et al. In this manuscript, in particular, we assume that the instantaneous firing rate of a neuron is a function of its membrane potential, and we use ensemble averaging to obtain the average firing rate of a homogeneous population of independent units.

For the transfer function, a common choice for population models is a threshold-linear non-linearity Wilson and Cowan, ; Ledoux and Brunel, Such models have been used, for example, for the investigation of orientation selectivity in the early visual system Carandini and Ringach, ; Ernst et al.

Emergent Collective Properties, Networks and Information in Biology

Recently, a theory for the analytic derivation of transfer functions in the case of leaky integrate-and-fire neurons has been proposed Ostojic and Brunel, In our manuscript, we chose an exponential transfer function Jolivet et al. Both the reset and refractoriness are incorporated in the self-inhibition of neurons. The rate equations arising from this choice are of Lotka-Volterra type and have already been used in modeling neural activity McCarley and Hobson, ; Fukai and Tanaka, ; Billock et al.

In some cases, Lotka-Volterra equations display the same attractor landscape as linear integrators complemented by a threshold-linear non-linearity Hahnloser et al. An important feature of our method is the correspondence between stability properties of the rate equation and of the underlying point process model. This correspondence goes far beyond the matching of attractive fixed points, and includes more complex attractors and associated bifurcations. The main aim of the present work is to demonstrate the correspondence between the point process model and the associated rate equations in different situations.

We study feed-forward networks, but also recurrent networks both in the case of single attractive or repulsive fixed points, as well as in the multistable case. Finally, we show that stable limit cycles are also correctly mapped from the rate equation to the point process model.

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Here, we briefly recapitulate the model introduced in Cardanobile and Rotter In this model, a neuronal population is characterized by its mean membrane potential V t. The mean membrane potential is assumed to obey a perfect integrator dynamics Tuckwell, At this step, we have assumed that the populations are statistically homogeneous in the sense that the distribution of synaptic strengths from neurons of population j to neurons in population i is narrow.

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Action potentials are emitted according to a stochastic mechanism, as in escape noise models for single neuron dynamics Gerstner and Kistler, The interpretation of escape noise models is the following: the membrane potential as well as the spike threshold are subject to spontaneous fluctuations on very short time scales. These fluctuations are both due to internal noise on the level of synapses and on the level of membrane and to fluctuating input. This transfer function depends on the state of both the neuron and the network. A good choice for the transfer function for neurons is an exponential function Rotter, ; Carandini, ; Jolivet et al.

We have verified through simulations data not shown that this choice is also suitable for populations of integrate-and-fire neurons. Inserting Eq. We refer to them as the rate equations of the system. They reflect the ensemble behavior of a homogeneous population of spiking neurons. We will use these equations exactly with this interpretation in mind. We would like to stress that the same type of dynamic equations has previously been employed in the neuroscientific literature McCarley and Hobson, ; Fukai and Tanaka, ; Billock et al.

We use the term homogeneous loosely, without a specific statistical framework in mind. Although the exact matching of spiking units to rate equations is a very interesting issue, the focus of this paper is to show that the rate equations offer a precise description of the expected behavior of multiplicatively interacting point process also in the time-dependent regime and to exploit the possibilities offered by this description.

In our firing rate equation, we have disregarded the leak term in the voltage dynamics. The question of whether leak terms can be consistently included into the equation is subject of current research. The focus of the present work is to demonstrate with neuroscientifically relevant examples that the mapping from the spiking model to Lotka-Volterra equations holds for all stable regimes. The examples included here are multistable systems, and systems with stable limit cycles. We have verified the mapping also for chaotic attractors and Hopf bifurcations, but we have excluded these examples from the present manuscript for the sake of readability.

The arguments given in this and our previous manuscript, and the examples of microcircuit design discussed here demonstrate unambiguously that the framework of Lotka-Volterra equations can indeed establish a solid connection between spiking dynamics of neural networks and their mean-field description. All simulations of networks of spiking neurons were implemented in the programming language Python van Rossum, , the scripts are available upon request. We employed time-driven solvers, based on a fixed step size.

Time steps were chosen between 0. The goal was to keep the probability of missing a spike as small as possible. In all cases, we have checked that our results are robust against a further decrease of step size data not shown. For the simulations, we resorted to a natural reformulation of the model in discrete time. Due to the exponential transfer function, firing rates couple multiplicatively to incoming spikes such that. Simulations are quite effective, since only the state variables of those neurons must be updated that receive spikes. In addition, Python supports vectorized computations of using the numpy Oliphant, module.

For the for the estimation of the probability of observing an unstable network Figure 4 and for the simulation of the spiking central pattern generator Figure 5 we used an event-based solver for increased precision, based on the Doob-Gillespie algorithm Doob, ; Gillespie, A specific strength of our approach is the possibility of determining the stationary response rate analytically, for a constant stimulus.

This population model describes a set of unconnected, independently firing neurons. Equivalently, it can be interpreted as the expected behavior of a single neuron, inferred from multiple independent observations trials. This arrangement typically leads to an output spike train that is slightly more regular than a Poisson process Softky and Koch, ; Shadlen and Newsome, , and that has weak negative serial correlations between adjacent inter-spike intervals Nawrot et al. The two fixed points, which characterize the stationary solutions of Eq. It is easy to verify that the more positive of the two fixed points is always globally attractive, and the other one is always globally repelling.

Apart from this simple non-linear response to stationary inputs, a feed-forward multiplicative unit also exhibits a power-law transient in its relaxation behavior, see Figure 2. Power-law relaxation to equilibrium has been described in real neurons Lundstrom et al. In this paper fractional differentiation was used to model neuronal integration properties, with the aim describe slow time-scale dynamic properties.

Figure 2. Feed-forward populations. Left panel: schematic of a two-population feed-forward circuit. Solid line marks the power-law region. If a unit has more than one input population, its total signed input is the sum of all input rates, weighted by the corresponding coupling constants. This suggests that the equilibrium dynamics of a multi-layer feed-forward network can be obtained by the following simple step-by-step procedure: first, one computes the equilibrium firing rates of the populations at the lowest level, induced by their respective inputs.

The equilibrium rates of populations at higher levels are then obtained by the same method, treating the activity of all lower level populations as input. As an example to illustrate the potential of such simple feed-forward networks, we demonstrate how to implement a system that is selective for inputs of strength within a prescribed range. This can be achieved with an arrangement of four populations, as depicted in Figure 3.

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Following the algorithm explained above, the output rate is given by. Figure 3. Band-pass filter. Population 3 modulates the activity to adjust both the width and the slope of the filter. Center panel: the theoretical equilibrium response light gray was computed by determining the fixed points of Eq. Simulations were performed by imposing Poissonian spike trains with different rates for the input population. The axes indicate input rate vs.

Right panel: normalized histograms of output rates measured during multiple trials, for three different input levels: onset green , midpoint between onset and peak red , peak blue of the tuning curve.


The area under the ROC curve is always greater than 0. So, if the term in the brackets is positive, the output activity will increase for increasing input, whereas if it is negative, the output activity will be 0 for strong inputs. Since we want to construct a network selective for a certain range of inputs, we impose that. On the other hand, given that the input vanishes for high rates, the output is maximal at the point where the input rate to unit 3 overcomes the global inhibitory signal to unit 3, i. So we obtain a bound on the global inhibition level given by. Under the conditions Eq.

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An example for a specific set of network parameters is given in Figure 3. All computations in this network are performed on the level of mean firing rates of the neurons. Therefore, it is important to assess the behavior of the corresponding spiking network, e. Results are given in Figure 3 and show, using ROC analysis, that different inputs can be well distinguished based on spike counts that are extracted from observations of finite duration.

The situation for a recurrent network is typically more involved. The attractive regimes are not necessarily corresponding to stable fixed points. Depending on the number interacting units, competitive Lotka-Volterra systems can display a variety of different phenomena:.

Here, we focus on one specific question of both theoretical and practical relevance: do the stability properties predicted by the rate Eq. Similar investigations have, in fact, been carried out for networks of integrate-and-fire neurons Brunel, ; Mattia and Del Giudice, ; Ledoux and Brunel, as well as for binary neurons van Vreeswijk and Sompolinsky, , We obtain a condition similar to the saddle-node bifurcation caused by a rate instability in LIF networks Mattia and Del Giudice, ; Ledoux and Brunel, To start our investigation, we recall that a sufficient condition for the dynamic stability of a non-linear equation can be derived from the existence of a so-called Lyapunov function for the system.

Under certain conditions, such a function can be constructed for our networks. Specifically, if A r denotes the recurrent part of the coupling matrix, then the system is stable, if all eigenvalues of A r have a negative real part. With a slight abuse of terminology we will refer to such not necessarily symmetric matrices as negative definite. In this case, the total firing rate of all populations taken together is a Lyapunov function of the system, thus guaranteeing stability.

This gives a handy sufficient but not a necessary condition for bounded rates in the system Cardanobile and Rotter, In two-dimensional systems describing mixed networks of homogeneous excitatory and inhibitory populations, dynamic stability can be assessed independently of a Lyapunov function. For such a system, the conditions for stability can be stated exactly, here made plausible using a simple heuristic argument.

If the system was feed-forward and in equilibrium, the stationary rates of both populations would be and respectively. Inserting one expression into the other, we obtain. We tested numerically the validity of this heuristic condition. Figure 4 shows simulation results that confirm the heuristic arguments made here. The condition given here is, in fact, a dissipativity condition which guarantees that the rate decays to 0 in absence of external input.

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Accordingly, if external input is present, the condition guarantees that the expected rate settles to an equilibrium value. It is interesting to mention that a linear system with the same coupling matrix has an additional stability condition. In fact, the condition given in Eq. The system is stable if and only if the determinant is larger than 0. For linear systems, one needs that additional constraints that the trace must be negative, which is not needed here.

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Figure 4. Stability properties. Shown is a Monte Carlo experiment that illustrates the condition for dynamic network stability derived in the main text. If the total rate was small, the network was classified as stable, otherwise as unstable. For each network trials were performed. Central pattern generators have been studied on an abstract level by many authors, for a review see Rabinovich et al. Lotka-Volterra type equations have also been used as specific models Venaille et al. However, no conclusive argument has been given why these systems would describe the mean-field dynamics of a neuronal network.

In our framework, in contrast, using Eq. The main goal of this section is to demonstrate that the spiking networks behave according to the predictions of the rate equations also the latter have stable limit cycles, rather than attractive fixed points. There are two fundamental alternatives to generate periodic patterns with Lotka-Volterra equations.

Already the classical two-dimensional Lotka-Volterra system used in population dynamics predicts periodic orbits under certain conditions. More specifically, in that case a whole family of periodic orbits exists, surrounding a stable fixed point. This can be proven by using the fact that the system possesses a constant of motion with closed contour lines. Thus, due to intrinsic noise of the spiking system, these oscillations will not generate periodic patterns with a stable, precise frequency.

Multiplicative point processes indeed exhibit periodic activity patterns with drifting frequencies, when connections are chosen according to the classical Lotka-Volterra system. However, we do not address this type of oscillator in this work. Alternatively, it has been observed that higher dimensional Lotka-Volterra systems can have stable limit cycles May and Leonard, In this case, the noise produced by the system is counterbalanced by the attractivity of the limit cycle.

As a consequence, such oscillations have a stable frequency. Furthermore, since the limit cycle is attractive, the spiking network described by the rate equation is attracted to a periodic activity regime.