DYNAMIC INFORMATION ROUTING IN NEURONAL CIRCUITS
Jorge Medina Hernández
Master’s Degree in Physics of Complex Systems (Specialization/Pathway in Complex Systems) Centre for Postgraduate Studies
Academic Year 2019/2020
NEURONAL CIRCUITS
Jorge Medina Hernández
Master’s Thesis
Centre for Postgraduate Studies University of the Balearic Islands
Academic Year 2019/2020
Keywords:
Neuroscience, Information routing, Neuronal circuits
Thesis Main Supervisor’s Name: Dr. Claudio Mirasso Santos Thesis Co-supervisor’s Name: Dr. Víctor Martínez Eguíluz
To Claudio and Víctor, for their help and dedication throughout this project despite the current circumstances.
To the best flatmates in the world Ana, Jorge, Javi and Sose for this great year.
To my mother, father and sister for their unconditional love and support.
The emergence of flexible channels of information in brain networks is a fundamental issue in neuroscience. Several topological properties have been suggested to define highly influential nodes, the most important being the "hub nodes", that is, nodes (brain regions) with a disproportionate number of connections to other parts of the brain. Nevertheless, the influence of a node does not only depend on its degree and position in the network, but on its internal dynamics. Recently, it has been observed that nodes with an oscillating frequency higher than those of their neighbors can act as functional hubs, redirecting the information. Thus, in this work the dynamics of a simple neural network have been analyzed to better understand which is the mechanism that transforms a normal node into a highly influential node. Specifically, we have focused on the propagation of signals in chain networks whose evolution is determined by the Hodgkin-Huxley model. First, we have studied the propagation of sinusoidal signals applied to a high-frequency neuron, verifying the existence of a certain interval of oscillating frequencies that allows the transmission of information across the network, in agreement with the literature. This interval was found to be linked to the capacity of the high-frequency neuron to set optimal relative spiking times of the other neurons in the network. Then, we considered the competition of two signals when they were applied to a high-frequency neuron and an ordinary neuron, as well as two high-frequency neurons. The results indicate that both the oscillating frequency of the neurons and the frequencies of the input signals determine the information transmission.
Acknowledgements i
Abstract ii
1 Introduction 1
1.1 Biological description of the neuron . . . 1
1.1.1 Basic structure . . . 1
1.1.2 Membrane potential at rest . . . 2
1.1.3 Nernst potential . . . 3
1.1.4 The action potential . . . 3
1.1.5 The synapse . . . 6
1.2 The model . . . 8
1.2.1 The Hodgkin-Huxley model . . . 8
1.2.2 Synaptic coupling . . . 11
2 State of the art 13 2.1 Information routing . . . 13
2.2 Related work . . . 15
3 Methodology 17 3.1 Methods . . . 17
3.1.1 Mean firing rate . . . 17
3.1.2 Cross-covariance . . . 18
3.2 Procedure . . . 18
4 Results 20 4.1 Phase locking and rate modulation . . . 20
4.2 Propagation of a single signal . . . 24
4.2.1 Towards the stationary cross-covariances. . . 24
4.2.2 The propagating regime . . . 26
4.2.3 Propagation of signals with high amplitude . . . 29
4.2.4 Dependence with the input frequency . . . 31
4.3 Propagation of two signals . . . 32
4.3.1 Single inhomogeneity . . . 32
4.3.2 Coexistence of two inhomogeneities . . . 42
Bibliography 47
1 Scheme of a neuron . . . 2
2 Sodium and potassium channels . . . 4
3 Phases and propagation of an action potential . . . 5
4 Diagram of a chemical synapse . . . 6
5 Diagram for the Hodgkin-Huxley model . . . 8
6 Dynamical routing. . . 14
7 Network topologies . . . 16
8 Phase locking . . . 21
9 Relative phases between adjacent neurons . . . 22
10 Gain and phase shift . . . 23
11 Stationary cross-covariances . . . 24
12 Stationary cross-covariances 51 neurons . . . 26
13 Traces of the potentials for different inhomogeneities . . . 27
14 Dependence of the response with the amplitude of the signal and the conductance . . . 29
15 Evolution of the potentials for high-amplitude signals . . . 30
16 Firing rates and Fourier spectra for high-amplitude signals . . . 30
17 Response for different signal frequencies. . . 31
18 Response: signals placed at the extrema . . . 32
19 Mean firing rate. Signals placed at the extrema . . . 32
20 Traces of the potentials. Signals placed at the extrema . . . 33
21 Transient phase towards the phase locking regime . . . 34
22 Amplitude modulation. Signals placed at the extrema . . . 35
23 Response. One signal in the middle . . . 36
24 Mean firing rate. One signal in the middle. . . 37
25 Amplitude modulation. One signal in the middle. . . 37
26 Firing rates comparison. One signal in the middle . . . 38
27 Response for different frequencies of the signals. One signal placed in the middle.. . . 39
28 Response. Signals placed in adjacent neurons. . . 41
29 Cross-covariances: two detunings. . . 42
30 Position of the drops in the cross-covariances . . . 43
31 Signal competition when there are two HFNs . . . 44
1 Synapse classification . . . 7
Abstract
In this chapter we present the main features of the neurons and focus on the coupling among them.
We begin by explaining the basics of the most relevant aspects for the neuron-neuron interaction from a biological point of view, such as the concepts of membrane potential, action potential and synapse. Then, we introduce the model used for the simulations, which is essentially the Hodgkin-Huxley, a model designed with experimental support and able to reproduce the known types of neural spiking activity.
1.1 Biological description of the neuron 1.1.1 Basic structure
The central nervous system (CNS), constituted mainly by the brain and the spinal cord, is a highly complex structure present on most of the animal kingdom, responsible for the external and internal information processing and the coordination of all the different parts of the body. At the cellular level, its components can be splitted between neurons and glia.
On the one hand, neurons are electrically excitable cells able to communicate with other cells by producing electrical impulses. Then, the glia encompasses all the different cells in the CNS that do not produce electrical impulses, but provide support for the neurons by maintaining adequate physicochemical conditions (homeostasis). Thus, although the glia can have an impact on the information transmission [Fol+08], we will focus on the cells mainly in charge of such task: the neurons.
Figure1shows the basic structure of a neuron. It contains the cell body (soma) and several terminations. Usually, the output terminals in the long termination (axon) propagate the electric impulse to the smaller terminations (dendrites) of other neurons, although it can be coupled to other parts of the cell. Thus, typically dendrites can be thought of as input terminals and the ramifications of the axon as output terminals.
Figure 1. Scheme of a neuron. The cell body (soma) has several small terminations (dendrites) from which the neuron usually receives electric impulses originated by other neurons; and a large projection (axon) that conducts the pulses to the output terminal. The Schwann cells, drawn in orange, are attached to the axon and form the myelin sheath, which covers the membrane and improves the propagation of the electric impulse [Dom18].
1.1.2 Membrane potential at rest
Since the cell membrane plays an important role in the transmission of the electric impulse, we will now analyze its structure. Similarly to other cell membranes, the neural membrane is a semi-permeable, lipid bilayer separating the interior of the cell (cytoplasm) and the outside environment (the extracellular medium), both having diluted ions embedded. Thus, as any other cell it will obey the following principles at resting conditions:
1. Equimolality: The concentration of osmotically active particles at both sides of the membrane has to be roughly the same.
2. Charge neutrality: The number of extracellular anions and cations has to be roughly the same. The same relation is verified among intracellular anions and cations.
These principles arise from the balance of forces when the equilibrium is reached: the osmotic pressure in the first case and the electrostatic forces in the second one. However, in reality there is a small asymmetry in the ion concentrations, of the order of picomols (10−12 mol), which leads to a non-zero resting potentialV0 of the order of−70 mV. In particular, for the most relevant ions in the generation of the action potential we have
[Na+]o>[Na+]i, [K+]o <[K+]i, (1.1) whereoandirefer to variables outside and inside the cell, respectively. Hence, taking into account the value ofV0and (1.1), we notice that both the chemical gradient (CG) and the electrostatic gradient (EG) are directed towards the cell in the case ofNa+, while forK+
the CG is directed outwards and the EG inwards. Then, why does not theNa+follow both gradients and reach the electrochemical equilibrium? The answer lies in the capacity of the membrane to actively transport theNa+ions outside the cell. The mechanism is known as the sodium-potassium pump, which transports threeNa+ atoms outside and twoK+ atoms to the inside at the expense of the hydrolysis of an ATP molecule, which provides the necessary energy for beating the gradients.
1.1.3 Nernst potential
If we now consider that only a certain ionAcan flow and no other forces rather than the chemical and electrostatic ones are present, the potential at which the ionic concentrations reach equilibrium is given by
EA= RT
zF log [A]o [A]i
, (1.2)
whereR = 8.32 Jmol−1K−1 is the universal constant of the gases,T is the temperature, zthe valence and charge of the ion andF = 96500 C/molis the Faraday constant. This equilibrium potential is also called theNernst potential orreversal potential for ionA.
Since the restriction of only one ion being able to cross the membrane does not hold in reality, we can not use the previous equation to calculate the potential at restV0. However, it is still a relevant quantity as it sets the directionality of the ionic currents across the membrane:IAtowards the inside whenEA> V and vice versa, beingV the membrane potential.
1.1.4 The action potential
The generation and transmission of electric impulses is a unique feature of the neurons that allows them to communicate with other cells. Specifically, these impulses are electrical transmembrane currents linked to the propagation of anaction potentialorspike- an abrupt and transient change of the membrane voltage. In order to understand the underlying mechanism behind this phenomenon, we must first analyze in greater depth the ionic flow across the membrane. Aside from the active transport of theNa+andK+ions, there are ion channels distributed along the membrane which do not need an energy input to allow the ion flow. Although they might be permeable to the passage of more than one type of ion, they are generally ion-specific and their state is linked to the membrane voltage.
Specifically, channels can be open, closed or inactivated as a consequence of the disposition of theirgates- protein segments whose spatial disposition depends on the voltage.
Figure 2. Sodium and potassium channels of a neural membrane, functionally constituted by four voltage-dependent gates. TheK+ channel has fournactivation gates, while the Na+channel has threemactivation gates and an inactivation gateh[Gro].
In figure2a scheme of theNa+andK+channels is shown. The first one has functionally threem activation gates and one inactivation gate h, while the second one has four n activation gates. As we will now explain, the gating configuration of these channels will set the dynamics of the ion flow. The generation of the action potential (Fig.3a) can be splitted in the following phases:
1. Initial stimulus. If a depolarizing stimulus, smaller than a certain threshold, is applied to the membrane, the voltage-gatedNa+channels do not activate and due to the active transport the membrane potential is restored to the resting state. However, if the stimulus is larger than the threshold the opposite situation is verified, allowing the flow ofNa+ions towards the inside of the cell, following the electrochemical gradient.
2. Depolarization. The entry of the Na+ further depolarizes the cell. This process continues until the change in the voltage leads to the inactivation of theNa+channels, with thehgate blocking the ion flow; and the opening of theK+ channels which produces a hyperpolarizing current of K+ ions moving towards the extracellular environment. Thus, at the peak of the depolarization we would approximately have IK+ =−INa+.
3. Repolarization. With progressively more sodium channels blocked and potassium channels opened, theIK+ is able to hyperpolarize the membrane.
4. Refractory period. TheK+channels gradually close, but there is an excess ofK+ flowing outside, leaving the membrane temporarily hyperpolarized with respect to the resting state. During this phase theNa+channels are mostly blocked, as theh gate takes some time to stop blocking the entry. As a consequence, along this period a hypothetical depolarization phase can not take place and the generation of action potentials is disabled.
(a) (b)
Figure 3.(a) Scheme of an action potential. When the stimulus is larger than the threshold the sodium channels open, causing a fast depolarization. Then, as theNa+ channels become inactivated, theK+channels open and gradually restore the membrane potential (repolarization), ending with a transient hyperpolarization (refractory period) [Gup+16].
(b) Propagation of the action potential. When the entry ofNa+ions occurs, the neighboring area begins to depolarize due to the flow of charges in the direction parallel to the membrane.
Then, when the depolarization reaches the threshold an action potential is generated, effectively propagating it along the membrane [Oli11].
Then, due to the structure of the membrane and the configuration of the charges, an action potential can effectively propagate across the membrane. During the depolarization phase, the flow of positiveNa+ions towards the cytoplasm leaves local regions inside and outside the membrane positively and negatively charged with respect to its surroundings (see figure 3b). Hence, lateral, local currents on both sides of the membrane will take place in order to balance the difference in charges, which in turn will depolarize the neighboring areas.
Subsequently, when the depolarization in these adjacent regions reaches the threshold, an action potential is generated, effectively propagating the action potential. This whole process involving the action potential and its transmission across the membrane is often referred to asnerve impulseand is crucial for neural communication and coordination, as we will see in the following section. In laboratory conditions, where a stimulus can be induced in arbitrary locations, the action potential can propagate bidirectionally; although in reality the stimuli are mostly applied in the dendrites or the soma and the transmission is directed towards the axon terminals.
1.1.5 The synapse
Although neurons can interact with other neurons, muscular cells and glandular cells through a directed coupling known assynapse, we will focus specifically on the neuron- neuron interaction, usually between the axon of the sending neuron and the dendrites of the receiving one. Fundamentally, we can distinguish between electrical and chemical synapses, independently of the target cell. The first one is the simplest case, in which the cytoplasms of the signal-passing neuron (presynaptic neuron) and the target (postsynaptic) cell are connected by special channels called gap junctions. This allows a direct transfer of ions, rapidly altering the voltage of the postsynaptic cell. Due to the nature of the coupling, the interactions are mostly bidirectional and the received signal can only be equal or smaller than the one originated at the presynaptic cell, i.e. they lack gain.
Figure 4.Simplified diagram of a typical chemical synapse between two neurons [Ion+20].
The action potential in the presynaptic neuron (yellow) activates theCa2+channels (red), which allows the release of the neurotransmitters embedded in vesicles towards the synaptic cleft. These chemical substances ultimately open or close the ion channels (pink) of the postsynaptic neuron (green), altering its membrane potential.
On the other hand, chemical synapses involve the release of chemical messengers known as neurotransmitters. It is the most frequent form of synaptic activity and allows the transmission of impulses in only one direction. The process, shown in figure4, can be splitted into several phases. First, an action potential generated in the presynaptic neuron reaches the axon terminal, causing the opening of theCa2+channels due to the electrical depolarization of the membrane. Subsequently,Ca2+ ions flow inside the terminal and change the shape of calcium-sensitive proteins attached to the synaptic vesicles containing the neurotransmitters, which make them fuse with the membrane of the presynaptic cell.
In the aftermath, the neurotransmitters are dumped into thesynaptic cleft- a gap of around
20-40 nm between the presynaptic and the postsynaptic terminals, an order of magnitude higher than the separation in the gap junctions required for the electrical synapses. Even though some of them diffuse within the extracellular environment, a significant proportion binds to chemical receptors in the membrane of the postsynaptic cell and cause, directly (ionotropic receptor) or through a metabolic pathway (metabotropic receptor), the opening of specific ion channels. This is a key step in the process, as it is the one in which the voltage of the postsynaptic cell is affected. Lastly, due to thermal vibration, neurotransmitters split from the receptors and diffuse away, allowing the postsynaptic neuron to return to its resting state, and are ultimately reabsorbed by the presynaptic terminal or broken down by enzymes (biological catalytic molecules).
Depending on the effect produced on the postsynaptic neuron, we can classify the change in the membrane potential as an excitatory postsynaptic potential (EPSP) or an inhibitory postsynaptic potential (IPSP). EPSPs make the postsynaptic neuron more likely to fire its own action potential, generally by inducing its depolarization, while IPSPs are usually hyperpolarizing and make the target neuron less prone to trigger an action potential.
Furthermore, since the shift in the membrane voltage depends on the type of ion channels activated, which in turn is a consequence of the messengers dumped into the synaptic cleft, chemical synapses can be classified according to neurotransmitter released (see table1).
Synaptic
type Neurotransmitter Brief description Main receptors
Glutamatergic glutamate Most abundant excitatory neurotransmitter in vertebrates
NMDA, Kainate and AMPA (ionotropic);
mGluR receptors (metabotropic) GABAergic
γ- aminobutyric
acid
Main inhibitory neurotransmitter in mammals
GABAA(ionotropic) and GABAB(metabotropic)
Cholinergic
acetylcholine and butyryl-
choline
Main neurotransmitter used by the parasympathetic nervous system
Nicotinic receptors (ionotropic) and muscarinic receptors
(metabotropic) Adrenergic
norepinephrine and epinephrine
Main neurotransmitter used to stimulate the sympathetic nervous
system
αandβreceptors (metabotropic)
Table 1. Synapse classification according to the released neurotransmitter.
In practice, most numerical models rely on glutamatergic and GABAergic ionotropic currents to simulate the synaptic activity. Specifically, NMDA and AMPA are often used for glutamatergic synapses and GABAAfor those of the GABAergic type.
1.2 The model
1.2.1 The Hodgkin-Huxley model
Hodgkin and Huxley [HH52] performed a series of experiments on the giant axon of the squid in order to understand the initiation and propagation of action potentials; and found three ionic currents involved: the sodium current, the potassium current and a leak current mainly formed byCl−ions. Then, with the support of the experimental data, they designed a model for the evolution of the membrane potential.
Figure 5. Diagram for the Hodgkin-Huxley model. The scheme for the neural membrane is shown on the left and its equivalent circuit on the right. The diagonal arrow across the diagram of a given resistor indicates that the resistance is not fixed, but changes depending on whether the ion channel is open or closed [Ger+14].
Essentially, they modeled the flow of ions across the membrane as an electric circuit, depicted in figure5. The semi-permeable membrane is represented as a capacitor with capacitance C and I refers to the input current injected into the cell, which sets the neuron’sintrinsic frequency, i.e. the firing rate it would have if there were no coupling with other neurons. Then, both theNa+andK+ionic channels are modeled as resistors with resistancesRNaandRK, which depend on voltage; and the electrochemical gradients arising from the difference in ion concentrations inside and outside the membrane are represented by batteriesENaandEK, whose voltages correspond to the reversal potentials (1.2) for the ions. Lastly, the unspecific ion flow is modeled by the constant leaking resistanceRLand a reversal potentialEL. Thus, applying Kirchhoff’s first law in nodea we obtain
I(t) = IC(t) +INa(V, t) +IK(V, t) +IL(t). (1.3) Substituting the current on the capacitorIC =CV˙ and writing the previous equation in term of the conductancesgi =R−1i we get
CdV
dt =I(t)−GNa(V, t)(V −ENa)−GK(V, t)(V −EK)−GL(V −EL), (1.4)
whereV −Eiis the total voltage across channels of typei. Then, each conductanceGi is assumed to be the sum of the conductances across the open channels of ioni, i.e.
Gi(V, t) =
Ntotal
X
c=1
¯
gi =Nopen(V, t) ¯gi = ¯GiPi(V, t), i= Na,K, (1.5)
beingg¯i the conductance of an individual channel,G¯i = ¯giNtotalthe maximal conductance andPi =Nopen/Ntotalthe fraction of open channels, or equivalently the open probability.
As we discussed in section1.1.4, the opening of a channel depends on the state of its activation and inactivation gates. Thus, assuming that gates of the same kind are identical and all gates act independently, for a channel of ion i with ki activation gates and li
inactivation gates the opening probability is given by
Pi(V, t) =akibli, i= Na,K, (1.6) wherea(V, t)andb(V, t)are the probabilities of having an activation subunit opened and an inactivation subunit not blocking the channel, respectively. By fitting the experimental data Hodgkin and Huxley determinedkK= 4,lK= 0,kNa= 3andlNa= 1; and denoted byn andmthe open probability of theK+andNa+activation gates and byhthe probability of theNa+inactivation gate not being blocking the channel. Although for instancekK = 4is consistent with the four-subunit structure of theK+channel, it is important to recall that these parameters have been chosen to fit the data and should be interpreted as functional definitions rather than models of the realistic ionic channels. Thus, even though theNa+ channel also has a four-subunit structure, the activation of the channel functionally operates as three independent gates [MYT98].
The dynamics of each gate is described using a simple kinetic scheme closed αβx(V)
x(V) open, x=n, m, h, (1.7)
such that the transitions closed→open and open→closed occur at voltage-dependent ratesαx(V)andβx(V), respectively. Thus, the probability of a gate opening in a small time intervaldtis proportional to the probability of having the gate closed (1−x), multiplied by the opening rateαx(V). Similarly, the probability of a gate closing in an intervaldtis proportional to the probability of having the gate open (x), multiplied by the closing rate βx(V). Therefore, the change in the open probability for a given gate will be given by the difference of these two contributions:
dx
dt =αx(V)(1−x)−βx(V)x, x=n, m, h. (1.8)
An initial guess for the rate functions can be performed based on thermodynamic arguments.
Since the transitions are associated with the movement of charged components of the gate across part of the membrane, they are likely to be limited by barriers requiring thermal energy, whose height should be affected by the membrane potential. Therefore, a transition requiring the displacement of an effective chargeqDxthrough the potentialV will need an energyqDxV; and the probability of having thermal fluctuations with enough energy to overcome this barrier will be proportional to the Boltzmann factorexp(−qDxV /kBT).
Hence, we would expect the rates to be of the form
αx(V) =Axexp(−DαxV /VT), βx(V) =Bxexp(−DβxV /VT), (1.9) whereVT =kBT /q. However, since these arguments rely on simplistic assumptions, in reality the transition rates will slightly differ from the initial guess (1.9) and are determined experimentally. Thus, substituting (1.5) and (1.6) back in (1.4) and adding the evolution equation for the gates (1.8), we arrive to the Hodgkin-Huxley model. Additionally, we will add the effects of the incoming synaptic currentsIsynand consider the inclusion of a signal currentJ(t)to analyze the propagation of information along the network. Hence, the extended model is given by
CdV
dt = ¯GNam3h(ENa−V) + ¯GKn4(EK−V) +GL(EL−V) +I+J+X Isyn, dx
dt =αx(V)(1−x)−βx(V)x. x=n, m, h,
(1.10) Without losing generality, we chose the following parameters [Mat14]:
C = 9π µF, ENa= 115 mV, EK=−12 mV, EL= 10.6 mV, G¯Na= 1080πmS, G¯K= 324πmS, GL= 2.7πmS, I = 340 pA,
(1.11)
being all the voltages of the model expressed relative to the resting potential. We have
used the experimentally determined transition rates αn(V) = 10−V
100 (e(10−V)/10−1), βn(V) = 0.125e−V /80, αm(V) = 25−V
10 (e(25−V)/10−1), βm(V) = 4e−V /18, (1.12) αh(V) = 0.07e−V /20, βh(V) = 1
(e(30−V)/10+ 1).
The model is able to reproduce the known forms of spiking activity of the neuron and has the advantage of being based on biological experiments, rather than assumptions over the dynamics. However, as can be noticed from (1.10) and (1.12), it is computationally costly, since the evolution of a single neuron requires the update of four variables (V, n, m, h) and the calculation of several exponential functions in each time step.
1.2.2 Synaptic coupling
We will only consider fast excitatory synaptic currents mediated by AMPA (A) receptors.
The simplest model approximates the dynamics by the two-state diagram [DMS98]
closed + T αβA
A open, (1.13)
beingαAandβAvoltage-independent transition rates andT the label for glutamate. Then, the fractionrj of open synaptic receptors in the postsynaptic neuronj is modeled by a first-order kinetic equation:
drj
dt =αA[T] (1−rj)−βArA. (1.14) Assuming all the reactions in the neurotransmitter release process are relatively fast and can be considered to be in steady state, the relationship between the glutamate concentration and the presynaptic voltageViis fitted to [DMS94]
[T] (Vi) = Tmax
1 + exp[−(Vi−Vp)/Kp], (1.15) where Tmax is the maximal glutamate concentration in the synaptic cleft, Kp sets the steepness of the sigmoid andVpcorresponds to the value ofVpreat which the function is half-activated. Thus, the synaptic current from neuronito neuronj is given by
Iij =gijrj(EA−Vj), (1.16)
whereVj is the postsynaptic potential,EAthe AMPA reversal potential andgij =g/kin(j) is the synaptic conductance, beinggthe maximal conductance andkin(j) the in-degree of neuronj. This definition ofgij ensures the sum of the incoming synaptic currents of each neuron has the same relevance for its evolution. Unless otherwise stated we have used the following parameters based on [DMS98] and [Mat14]:
αA= 1.1 mM−1ms−1, βA= 0.19 ms−1, Tmax = 1 mM, Kp = 5 mV, Vp = 62 mV, EA= 60 mV, g = 10 nS.
(1.17)
The rate constants αA and βA are known to depend on different factors and can vary significantly [Gei+97;HR97;KJ00]. Hence, the values chosen serve as a starting point to exemplify our results.
Abstract
In this chapter, a brief introduction to the idea of information routing is presented, a process of great importance in many technological and biological systems, especially in neuronal circuits.
After some basic definitions, the current state of the art based on experimental, theoretical and computational research is provided, ending with the work by Pariz et al. [Par+17]. The latter constitutes the most relevant research among those cited in the bibliography, since this project can be considered a continuation of it.
2.1 Information routing
The correct functioning of many complex systems depends on their ability for distributing and transferring information along its components. This is the case for artificial dynamic systems like the mobile communication network, but also for a wide number of biological systems, including the brain.
Technically speaking, the process of selecting a path for the flow of information in a network or across multiple networks is calledroutingand can be static or dynamic. On the one hand, static routing is a fixed pathway, implying that a signal applied on a certain node A will always reach those towards which there exists a path from A. In the context of the brain, the stimulation of a certain area would entail the activation of those other areas structurally connected to it, i.e. those reachable through successive synapses from the original region. However, in reality the information flow only reaches a fraction of the possible target regions, or in other words, the effective connectivity does not coincide with the structural connectivity [Bat+12]. The effective connectivity is flexible and depends on contexts and tasks, meaning that the pathway across which information can be directed changes according to the circumstances in a process known as dynamic routing (figure6).
Figure 6. Illustration of a dynamical routing problem. Sometimes it is necessary to route information from a sensor on the left to an effector on the same side (L-L connection in black). Other times the information must flow from left to right (L-R connection in blue).
The dynamic routing of information is the ability to change the direction of the information flow depending on the context [Thi].
An example of recent experimental evidence supporting this idea is the paper published in Science by Ciocchi et al. [Cio+15], demonstrating that the ventral hippocampus (vCA1) of rats sends different types of information - anxiety, goal-directed activity or task-related activity - to different areas of the brain depending on the context. Thus, it shows that vCA1 does not share the same information with all downstream brain structures, but performs a neural computation and transfers the information to specific regions. On the other hand, D.
Battaglia et al. [KTB16] have provided meaningful theoretical insights from the analysis of complex networks, focusing on those with oscillatory dynamics, widely observed in neural circuits. Specifically, researchers discovered that for every collective dynamic state of the full system, there is a certain pattern of how information is distributed and transferred between the system’s components. In other words, if the dynamic state changes qualitatively, the whole routing pattern is also modified, i.e. the routing pattern is dynamic.
Furthermore, recent findings in the field of computational neuroscience have suggested possible mechanisms underlying this phenomenon [Pal+17]. The conclusions arise from the analysis of a system of multiple circuits below the onset of developing oscillatory synchrony, which show short-lived and weakly synchronized collective oscillations with stochastically drifting frequencies; but when coupled together by long-range excitatory connections generate spontaneous bursts of synchrony affecting the whole system. In particular, during these high frequency (gamma) bursts, the drifting frequencies track each other and lead to transient phase-locking which can gate information flow, selectively enhancing or decreasing the information transfer along specific routes depending on the transient phase pattern.
2.2 Related work
Our proposal is to analyze the effects of high-frequency neurons in the propagation of signals and the establishment of a preferential direction for information flow. It is based on the work by Pariz et al. [Par+17] and can be considered a continuation of it. In the mentioned paper, researchers used the leaky integrate and fire (LIF) model described by
τidvi
dt =−(vi−vleak) +Ii+ηi(t) +Ji(t) +X
ij
Iij, Iij =aijgij
X
k
δ(t−tkj), (2.1)
where τi is the membrane time constant, vi the potential of neuroni; vleak is the leak membrane potential, theηiare independent Gaussian white noise currents,Ii is the base current,Ji(t)is the applied signal andIij are synaptic currents, beingaij elements of the adjacency matrix andtkj the time at which neuronj spiked. In this model, neurons fire when their potentials reach the thresholdvthr = 10 mV, after which are set tovres= 0 mV.
Such model is more efficient than the one designed by Hodgkin and Huxley (1.10) and allows for the simulation of larger neural networks, but does not account for all the spiking modes found in biological neurons. Researchers analyzed the propagation of sinusoidal signals (and non-periodical signals in the supplementary material) in networks ofN nodes, each being composed of 80 excitatory and 20 inhibitory neurons. For the task the most relevant methods used were the cross-covariance, Fourier transform and the time-delayed mutual information.
The first neural topology considered was a neural chain, where all the nodes had an intrinsic frequencyν¯except for the middle node, withν = ¯ν+ ∆ν (high frequency node or HFN).
The signal, applied onto the HFN, propagated across the network for a certain domain of positive inhomogeneities∆ν, with large values of the cross-covariances between the signal and the firing rates of the nodes, higher the proximity to the HFN. A similar behavior was seen in the amplitudes of the Fourier transforms at the frequency of the input signal for variable inhomogeneities∆ν, as the ordering of the sizes of the peaks labeled by∆ν coincided with the corresponding of the cross-covariances. Another finding related to the previous statement concerns the shape of the spiking rates of the nodes: the modulation of the firing rates of neurons with∆ν in the regime of good propagation resembled the original input signal, and appeared noisy otherwise. Lastly, the quality of the transmission was found to be significantly lower the higher the frequency of the signal.
Then, researchers studied the propagation of two signals, one applied to the HFN and the other to an arbitrary neuron. Interestingly, the signal from the HFN propagated across the
whole network, while the one in the ordinary neuron did only in the direction opposite to the HFN, i.e. the HFN had set a preferred direction for the transmission of the second signal. Thus, the cross-covariances between the firing rates and the second signal, as well as the Fourier amplitudes, showed high values in nodes located towards the optimal direction (away from the HFN) and vice versa.
(a) (b) (c)
Figure 7.Network topologies analyzed in [Par+17]: chain (a), circle (b) and a combination of a star and a chain (c). The HFN is plotted in yellow and the node where the second signal was applied in green. The chain analyzed in the article had 31 nodes and the signals were applied in node 20 (HFN) and 10 (ordinary node). The case (c) was also studied with the HFN and the ordinary neuron interchanged
Afterwards, the transmission across the network architectures shown in figure 7 was analyzed using time-delayed mutual information. In all the cases, the HFN set a preferred direction (away from the HFN) for the propagation of the signal applied to the ordinary node, while the signal from the HFN was transmitted across the full system. This was especially surprising in the fourth case, which corresponds to figure7c with the HFN and the ordinary neuron interchanged, i.e. the HFN placed in the chain and the ordinary neuron in the center of the star, conforming a structural hub due to its large number of connections.
Thus, as the signal from the HFN reached the whole network, it can be assured that its greater frequency turned the node into a functional hub, while despite the fact that the second signal was applied into a structural hub the signal transmission was not ensured, since it did not propagate across the chain. In other words, the intrinsic frequency of the nodes seemed to be a key feature for the establishment of the information routing pattern.
Finally, researchers focused on the 47 nodes CoCoMac connectome, an anatomical con- nectivity database of the macaque brain, to check whether the previous conclusions held when considering a biological network. As expected due to their verification in different topologies (figure7), the information flowed still from the HFN towards the whole network, even if it was not located in a structural hub, and the differences in the propagation when the HFN was placed in a hub were found to be only quantitative.
Abstract
The mean firing rate and the cross-covariances will be the basic tools for the analysis of the signal propagation. In this chapter we explain their calculation, as well as the course of action for the project and the motivation behind each step.
3.1 Methods
3.1.1 Mean firing rate
The Gaussian windows of width∆t=N h, beinghthe integration time step, is given by w[n] = exp −1
2
n−N/2 σw
2!
, 0≤n≤N σ≤0.5,
(3.1)
whereσw =σN/2is the standard deviation. In order to calculate the mean rate at timet, we apply a Gaussian window to the firing ratef(t)along the interval[t− ∆t2 , t+∆t2 ]and perform the average:
f(t) =¯ PN
n=0w[n]f(t− ∆t2 +n) PN
n=0w[n] . (3.2)
Therefore, the mean firing rate would be defined for the interval[∆t2 , T − ∆t2 ], beingT the integration time. We have takenσ = 0.3for all the simulations and widths∆t = 50ms and∆t = 4msfor the calculation of the cross-covariances and the peak of the Fourier transform atf0, respectively, except for those explicitly indicated.
3.1.2 Cross-covariance
The cross-covariance between two signalsx(t)andy(t)is given by
σxy(t) = F−1D h
FD(X, L)·
FD(Y, L)∗i
||X||||Y|| , (3.3) whereX(t)andY(t)are the rescaled variables
X(t) = x(t)−x(t),¯ Y(t) = y(t)−y(t),¯
(3.4)
andFD(X, L)is the discrete Fourier transform applied to the signal X padded withL zeros on both sides, beingLthe number of data points of the signalX. Since the signals must have equal length, during the calculation of the cross-covariance we must consider only the domain[∆t2 , T − ∆t2 ]for bothf(t)¯ andJ(t). Equation (3.3) represents the cross- covariances at different timest, i.e. the correlations betweenx(t0)andy(t0)when the latter is shifted a temporal intervalt: σxy(t) = σ x(t0), y(t0+t)
. Thus, the response shown in the figures corresponds to the maximum ofσxy(t).
3.2 Procedure
1. Propagation of one signal:
We will initially consider a chain-shaped neural circuit of 21 neurons, being the neuron in the center (11) the HFN with an intrinsic frequency displaced an amount
∆ν. For that configuration we will analyze the following features:
(a) Phase locking: According to the communication through coherence hypothesis [Fri01], neuronal oscillation locked at the appropriate phase may facilitate information transmission in the brain. Therefore, we would expect the ap- pearance of phase locking as a mechanism enhancing the signal propagation.
Since synaptic influence is maximized when the presynaptic neuron spikes immediately before the postsynaptic one, it could likely be in the form of delayed synchronization (DS), where the sender leads the receiver in time.
(b) Firing rate modulation: The input signal will produce a modulation of the firing rate in the HFN, and the transmission of information will cause the firing rates of the other neurons to resemble this modulation. By studying the relative sizes of the modulation (the gain), the potential limit for the propagation can be estimated. Additionally, the relative phase of the modulations can be also calculated.
(c) Stationary cross-covariances: We are interested in the transmission of informa- tion in the stationary regime, where the network has adapted to the incoming signal. In order to ensure the results obtained verify this condition, the time needed to reach stationary dynamics is computed. Afterwards, we will focus on the values of the cross-covariances to characterize the signal propagation.
(d) Propagating regime: As in the work by Pariz et al. [Par+17], the signals will propagate only for a certain interval [∆νmin,∆νmax]. In order to understand what causes the existence of such regime, we will analyze the traces of the potentials and the relative spiking times of the neurons.
(e) Dependence with the frequency of the input signal: We will analyze how sinusoidal signals propagate depending on their frequency. Thus, the results will determine if the system shows any specific behavior depending on the speed at which the signal varies, such as a low/high-pass filter nature, resonances or anti-resonances.
2. Propagation of two signals:
By including a second signal we can calculate the pattern of information routing and discuss which features affect the information propagation. We will evaluate the following configurations:
(a) Single inhomogeneity: One signal is applied to a HFN and the other one to an ordinary neuron. We will change the positions where the signals are applied and check if it has implications in the information transmission by measuring the cross-covariances.
(b) Coexistence of two inhomogeneities: As the intrinsic frequency is a key aspect for information routing and signals applied to HFNs propagate across the whole network [Par+17], in this part we want to inspect the possible outcomes arising from applying the two signals to HFNs with close detunings: |∆ν(HFNs)|=
|∆ν1−∆ν2| 1. Therefore, we will consider small variations in the values of∆ν(HFNs)and calculate the response of the network to the signals using the cross-covariances.
Abstract
The intrinsic frequency of the HFN sets the quality of the signal transmission. Thus, there is a specific frequency interval[∆νmin,∆νmax]out of which signals are not transmitted. Furthermore, the HFN is able to gate the information across the network, inhibiting or enhancing the propagation of signals depending on their directionality. Lastly, the transmission of information when we consider two HFNs is found to be dependent on both their intrinsic frequencies ∆ν and the frequencies of the applied signalsf0.
4.1 Phase locking and rate modulation
Figure8shows the evolution of the potentials and firing rates for the case where no signal is applied on the HFN (a,b) and the one where a sinusoidal signal is applied (c,d). In the first case, the network is able to reach phase locking, with all the neurons eventually synchronizing at the same spiking frequency. On the other hand, when a signal is applied the phases of the neurons remain relatively fixed as shown in figure9, oscillating at the frequency of the input signal around the constant phase shift obtained when no signal is applied. More importantly, these relative phases indicate that the HFN sets a clear ordering in the spiking times of the neurons, with neurons firing sooner the closer they are from the HFN. This fact will have important consequences in the propagation of the signal and will be analyzed in section4.2.2. Then, we see a clear asymmetry in the relative phase between neuron 11 (the HFN) and 12. Specifically, the distribution is elongated towards the zero phase shift, indicating that some times the lag between neuron 12 and 11 reduces more than expected. As it can be seen in figure8d, this fact is a consequence of the bigger gap between the minima of the firing rate of neuron 11 and 12, compared to the relative sizes of the maxima.
In addition, frequency locking occurs between neurons at the same distance from the HFN (see figure8d), placed in the center of the neural chain. This can be explained by the network connectivity and the features of the synaptic coupling. Since in a bidirectional chain network neurons only receive synaptic currents from the ones adjacent to it and only a fraction of the incoming current is transmitted in the outgoing one, there is a noticeable attenuation component. Thus, the amplitude of the firing rates oscillations at different distances from the HFN (i.e. different attenuation degree due to the intermediate synapses) will differ, preventing frequency locking.
(a) (b)
(c) (d)
Figure 8. (a) Evolution of the potentials of the HFN (neuron 11) with inhomogeneity∆ν = 4.16 Hzand the two nearest neurons at both sides, with neurons at equal distance from the HFN plotted in the same colors. Neurons at the right of the HFN are plotted in thin, solid lines and those at the left, in thicker and semi-transparent lines. (b) Firing rates for the neurons in (a). Figures (c), (d) are the corresponding of (a), (b) for the case where a sinusoidal signalJ(t) =Ascos(2πf0t)is applied, withAs= 2 pAandf0= 9 Hz. The simulated network is chain-shaped and contains 21 neurons, being the HFN placed in the center. The inhomogeneity was chosen to be∆ν= 4.16Hz, corresponding to an extra base current∆I = 80 pA.
(a)
(b)
Figure 9. Distributions of the relative phases between neuronsiandj, given by∆φi,j = φj−φi. The shape of the distributions can better be seen in the histograms of subfigure (b), while their width can better be estimated from the polar plot (a). The phase shift when no signal is applied is plotted as a black line. It remains locked and shows negligible oscillations, with maximum width of the order of10−2radians. The results for the other direction (11→1) are analogous, with neuron pairs at the same distance from the HFN showing similar phase shifts.
Specifically, considering the firing rate of neuroni fi to be given by
fi(t) = ¯fi+H(t,∆fi, φi), (4.1) whereHis a modulation of the firing rate, approximately of the same shape as the input signal for high transmission rates. In our case, in such situationH(t,∆fi, φi)would be a sinusoidal function of amplitude∆fi and phase φi. Furthermore, since the synaptic current is transmitted in a specific time window after their spike, there will be a delay in the propagation of the signal, which causes the firing rate oscillations to be shifted by a phase proportional to their distance to the source node. Both the attenuation of the firing rate modulation amplitude∆fi and the phase shift∆φcaused by the temporal delay are shown in Figure10.
We have obtained similar results for other chain lengths. Specifically, forN = 51neurons the attenuation coefficients plateau at0.12and keep fluctuating around that value, allowing the possibility of a successful propagation for longer chains.
(a) (b)
Figure 10. (a) Attenuation of the firing rate modulation amplitude∆f with the distance to the source node, i.e. the number of synaptic connections needed for the signal to arrive from the source node. (b) Relative phase between the firing rate modulation functions of the HFN and neurons at a given distance from it (blue). Both magnitudes have been calculated for an inhomogeneity
∆ν = 4.16 Hz, corresponding to a regime of maximum propagation. Although distances are discrete we have plotted the results using solid lines to gain a better intuition of the trend of each magnitude.
4.2 Propagation of a single signal
4.2.1 Towards the stationary cross-covariances
(a)
(b)
Figure 11. (a) Stationary cross-covariances between the firing rate of each neuron shown in the legend and the input signalJ(t) = 2 cos(2πf0t), withf0 = 9 Hz. The HFN (neuron 11) is plotted in red and we have plotted the other neurons in colors closer to blue the farther away from HFN. We have verified that neurons at the same distance from the HFN show similar cross-covariances, but those in the other direction (11→1) are not plotted for visualization purposes. (b) Convergence of the cross-covariances as a function of time for different inhomogeneities∆ν, averaged over several neurons placed evenly across the neural chain. The inner plot shows the convergence for several neurons averaged over the inhomogeneity values.
Figure11a shows the cross-covariances between the firing rate of several neurons and the input signal, which serves as a measurement of the response of the network to the signal. As expected, neurons closer to the HFN, where the signal is applied, show higher responses in general. Then, we notice the appearance of a specific inhomogeneity window
[∆νmin,∆νmax]≈[3 Hz,15 Hz], where the signal efficiently propagates. Thus, for large, small and negative detunings the signal only reaches neurons nearby the HFN. We have verified that neurons at the same distance from the HFN show the same cross-covariances, as expected from the symmetric configuration of the chain. However, very small differences were obtained (∼10−2) between both directions for the regions where the signal does not propagate, which are expected to disappear when averaged over realizations with varying initial conditions for the neurons. As we will discuss in section4.2.2, for that specific window the signal is not able to set a fixed spiking regime, where the ordering remains constant. This fixed spiking regime does not depend on the initial conditions and explains why for other detunings the difference in the cross-covariances for neurons at the same distance from the source is negligible.
Fig11b shows the convergence of the cross-covariances for different inhomogeneities and neurons in the main and inner plots, respectively. As it can be seen, the convergence is faster for the values of∆νwhere the response maximizes, reaching the stationary regime aroundt= 0.5s; while for other inhomogeneities the cross-covariances fluctuate near the stationary values for a significantly longer time. However, as we have mentioned before the cross-covariances for bad propagation regimes are not reliable and the presence or lack of their convergence do not provide meaningful information. We also notice a bigger dispersion in the cross-covariances for the initial times, whose explanation can be found in the inner plot. As expected, neurons closer to the source node adapt to the signal faster than those placed at higher distances, leading to a spread in the convergence for a fixed
∆νwhen averaging over neurons across the whole network.
Analyzing now figure 12, the equivalent of figure11 for a chain ofN = 51nodes, we conclude that the regime of non-negligible propagation approximately coincides, showing a decrease in the transmission in the high inhomogeneity region. This opens the possibility of successful signal propagation for longer chains, provided the detuning of the HFN lies within the optimal interval. Furthermore, comparing figures12b and 11b we see that within the high propagation regime the convergence is faster for higher values of
∆ν. In principle, a higher∆νincreases the locking frequency of the network in absence of any signal, i.e. rises the spiking rate of the whole network, which could result in a faster information transmission since it occurs in a specific time window after the action potentials, explaining the obtained result. In other words, as in a neural chain at each spike only the adjacent neurons can interact, if the spiking rate is higher the equilibrium is reached faster. As expected from the linearity of the network configuration, from figures 11and12we notice that the convergence time for any∆ν in the high propagation regime
seems to be approximately linear with the number of neuronsN.
(a)
(b)
Figure 12. (a) Stationary cross-covariances between the firing rate of each neuron shown in the legend and the input signalJ(t) = 2 cos(2πf0t), withf0 = 9 Hz. The HFN (neuron 26) is plotted in red and we have plotted the other neurons in colors closer to blue the farther away from HFN. We have verified that neurons at the same distance from the HFN show similar cross-covariances, but those in the other direction (11→1) are not plotted for visualization purposes. (b) Convergence of the cross-covariances as a function of time for different inhomogeneities∆ν, averaged over several neurons placed evenly across the neural chain. The inner plot shows the convergence for several neurons averaged over the inhomogeneity values.
4.2.2 The propagating regime
In figures11and12we have verified the existence of a specific range of inhomogeneities [∆νmin,∆νmax], slightly different in each case, for which the signal is propagated. We will now explain this phenomenon by analyzing the traces of the potentials for different values of∆ν, shown in figure13.
(a)
(b)
(c)
Figure 13.Traces of the potentials and the synaptic currents for∆ν = 0Hz(a),∆ν = 4.16Hz (b) and∆ν = 20.69Hz. The HFN was chosen to be neuron 3 in every case. The shape of the signal J(t)has been plotted in the upper part of each subfigure to better analyze the changes in the firing rates of the neurons.
It is well known that the effect produced by the postsynaptic current is maximum when the emitting neuron spikes immediately before the receiving one, and minimum if it does immediately afterwards, during the refractory period. Thus, each of the subfigures in figure13would in principle explain the observed behavior taking into account the relative spike-timing of the neurons:
(a) ∆ν ≤∆νmin:
If |∆ν|is sufficiently small, as in figure 13a, the input current is able to modify the firing rate of the HFN, changing its relative magnitude with respect to those of the adjacent neurons. As J(t)varies in time, whenJ > 0or J < 0the HFN progressively increases or decreases its firing rate, respectively. Therefore, there is a time interval where the presynaptic neuron always spikes before the postsynaptic one and the effect of the synaptic current is enhanced, allowing a high signal transmission;
alternated with another time window where the opposite situation takes place. The length of these time intervals would depend on the actual value of∆ν, being higher the favorable time window with positive values of∆ν and vice versa. For instance, if ∆ν 0 the input current would not be able to rise enough the firing rate of the source neuron to spike before the adjacent neurons and the signal propagation would be minimized. Nevertheless, if the amplitude of the signal is high enough to counterbalance the small detuning, signal propagation can take place, as shown in figure14c. This case will be analyzed in section4.2.3.
(b) ∆νmin ≤∆ν ≤∆νmax:
The traces of the potentials for this case are shown in figure 13b. The signal transmission is optimal, since always the presynaptic neuron spikes shortly before the postsynaptic one, favoring the propagation towards both ends of the network . It is important to notice that due to the attenuation delay in the transmission there can not be phase locking, as shown in figure8b, but the configuration is stable in the sense that the ordering of the spikes remains fixed.
(c) ∆ν > ∆νmax.
The relative spike-timing between the HFN and the adjacent neurons varies, being only optimal at specific times and sometimes the postsynaptic neuron spikes before the presynaptic one (see figure13c). As in case (a), this alternating behavior does not allow maximum signal transmission. Furthermore, as the relative spiking times highly fluctuate, the synaptic currents become noisy compared to the other cases.
If the HFN is receiving “randomized” synaptic currents, they could act as noise and explain why the cross-covariances are low even for the HFN (figure 11a, for
∆ν >15 Hz).
Lastly, we have calculated the responses for several values of the conductancegand the
signal amplitude As, shown in figure 14. For fixed As (b), the propagating regime is shifted to higher inhomogeneities for increasing values ofg; while comparing (c) and (d) we notice that the higher conductivity allows for better transmission of a wider range of signal amplitudes, being it specially restricted in the case of lowg (g = 5 nS). The size of the explored parameter space defined by∆ν, g, As and the cross-covarianceσf J
is quite narrow compared to its total volume, not allowing for a precise extrapolation to other regimes. Nevertheless, we can conclude that for the usual values of the conductivity g ∈ [5,10], there is a specific inhomogeneity range in the interval∆ν ∈[4.5,15] where low amplitude signals (As ∈[0,5]) can be transmitted.
(a) (b)
(c) (d)
Figure 14. (a) Graphical representation of the neural network (a), where the HFN (11) is plotted in red and the sample neuron (3) in green. (b) Response of the sample neuron for different inhomogeneities∆νand values of the synaptic conductanceg, taking an amplitude of the signal As= 2pA. (c)-(d) Response of the sample neuron for different inhomogeneities∆νand amplitudes of the signalAs, takingg= 5nSandg= 10nS, respectively. For the case where no signal is applied we obtain zero cross-covariances, but as soon as the signal amplitude increases the perturbation can propagate across the network.
4.2.3 Propagation of signals with high amplitude
We had seen that for negative detunings∆ν <0the neuron where the signal is applied alternates between spiking before and after the adjacent neurons, leading to a poor signal transmission. However, as it can be seen in figure14c, for high amplitude signalsAs≈ 30 pA, the signal propagates through the chain.
Figure 15. Evolution of the potentials (above) and synaptic currents (below) nearby the source node (neuron 11), forAs = 30 pA. The shape of the signalJ(t)has been plotted in the upper figure to better analyze the changes in the firing rates of the neurons.
As it can be seen in figure15, in the mentioned regime the strength of the input signalJ(t) is enough to temporarily shut down the source neuron, leading to alternated periods of fast spiking and subthreshold activity, following the frequency of the signal. Then, as the other neurons’ relative spiking time does mostly but not always favor the propagation towards the end of the chain; since during the shut-down period there is a loss of information, the resulting firing rates do not resemble the sinusoidal signal (figure16a). Nevertheless, the Fourier spectra still show clear peaks at the frequency of the input signal (figure16b). This fact along with the high cross-covariances seen in figure14c points out the propagation of relevant features of the signal across the network.
(a) (b)
Figure 16.(a) Mean firing rates of neurons nearby the end of the chain. (b) Fourier spectra of the mean firing rates. The frequency of the input signal isf0 = 7Hz.
4.2.4 Dependence with the input frequency
(a)
(b)
Figure 17.(a) Cross-covariance between the firing rate of neuron 10 and the signal applied to the HFN (3) for different values of its frequency. The smallest frequencyf0 = 1 Hzis plotted in blue, while the rest are plotted in colors closer to red the higher the frequency. (b) The equivalent for non-periodic signals of the formJ(t) = 2 cos(2πf0t) + cos(√
4πf0t) .
The frequency of the input signalf0can determine the quality of the propagation for certain inhomogeneities. As shown in figures17a and17b, the system generally behaves as a low- pass filter, favoring the propagation of slowly-varying signals over the fastly-varying ones (f0 ≥15 Hz). Nevertheless, there seems to be a narrow region with inhomogeneities∆ν, approximately between 6 and 11 Hz where all the periodic signals propagate, regardless of the frequency.
4.3 Propagation of two signals 4.3.1 Single inhomogeneity
Signals placed near the extrema of the chain
(a)
(b)
Figure 18. (a) Scheme of the network. The HFN (3) with ∆ν = 4.16 Hz is plotted in red, while the neuron with∆ν = 0where the second signal is applied (19) is plotted in yellow. (b) Cross-covariances between the firing rates of each neuron and the input currents transmitted to the HFNJ1(t)(blue) and the ordinary neuronJ2(t)(orange). Although neurons are discrete we have plotted the results using solid lines to gain a better intuition of the trend of the cross-covariances.
Figure 19. Evolution of the mean firing rate of the source node with∆ν = 0(neuron 19) and its surrounding neurons. The HFN is placed in the third position of the neural chain.
As shown in figures19and20a, during the first 500 milliseconds the whole network has not still reached phase locking and the situation nearby the second source node is analogous to the case of one signal with∆ν <∆νmin (figure13a). Hence, the spike timing of the source node alternates between optimal values and the worst case scenario, only allowing part of the signal to be transmitted to the adjacent neurons.
(a)
(b)
Figure 20.Evolution of the potentials nearby the source node with∆ν = 0(neuron 19) when the network has not reached phase locking (a) and when it has (b).
Then, when phase locking is achieved (figure20b) the situation is analogous to the good propagating regime (figure13b), with important differences. As in the mentioned case, neurons spike optimally for the transmission of the signal of the HFN, with presynaptic neurons spiking shortly before the postsynaptic ones. Thus, we see that this configuration partially fixes the ordinary source node (19) in a spiking regime where it spikes after neuron 18 and before neuron 20. Therefore, it will be in an optimal position to propagate towards
