article: The time scale of neuronal network dynamics is determined by synaptic interactions and neuronal signal integration, both of which occur on the time scale of milliseconds. Yet many behaviors like the generation of movements or vocalizations of sounds occur on the much slower time scale of seconds. Here we ask the question of how neuronal networks of the brain can support reliable behavior on this time scale. We argue that excitable neuronal assemblies with spike-frequency adaptation may serve as building blocks that can flexibly adjust the speed of execution of neural circuit function. We show in simulations that a chain of neuronal assemblies can propagate signals reliably, similar to the well-known synfire chain, but with the crucial difference that the propagation speed is slower and tunable to the behaviorally relevant range. Moreover we study a grid of excitable neuronal assemblies as a simplified model of the somatosensory barrel cortex of the mouse and demonstrate that various patterns of experimentally observed spatial activity propagation can be explained.
  Reliable propagation of activity is necessary for processing and transmitting sensory signals in the brain. During the last two decades, two prominent types of computational models have been studied to address this issue. First, the synfire chain consists of groups of spiking neurons connected in a feedforward architecture [1–4] potentially embedded in recurrent networks [5,6]. Second, rate propagation models [6–9] use a similar feedforward architecture, but instead of spikes they propagate fluctuations of the firing rate. In a synfire chain the refractory behavior of neurons after firing a spike is the crucial element of stable activity propagation [10]. The derivative of the membrane potential shapes spike density and sharpens the activity pulse [10,11]. Statistical methods [12,13] have been proposed for detecting the firing pattern of synfire chains in spike train recordings. Reliable neuronal firing patterns compatible with synfire chains have been observed in area HVC of the song-bird [14], while the statistical significance of synfire chains in cortical neurons is questionable [15]. Synfire chain models have been used for reproducing behavioral functions such as bird song generation [16] and monkey scribbling [17]. Systems of interacting synfire chains were also used for building a large-scale model of cortex [18,19]. While both synfire chain and rate propagation succeeded to model fast behaviors (behaviors on the order of milliseconds), they are expensive in terms of neuron numbers and therefore not suitable for reproducing behavioral phenomena that need a slower, and sometimes tunable, speed of activity propagation. Essentially synfire chains implement a clock, set by the delay of spike propagation and the rise time of excitatory postsynaptic potentials on the millisecond time scale [20,21]. In order to address this issue, we propose an excitation chain model, for activity propagation in a bidirectional chain of neuronal assemblies. Our model can be considered as a spiking version of excitable media [22–26], with an explicit link to the neuronal time scale of spikes. In contrast to previous synfire chain and rate propagation models, our model does not require an explicit feedforward architecture. Specific feedforward structures have not been observed in experiments in the neocortex so far. Here we propose an excitation chain model that is consistent with the following experimental connectivity data: first, inter- and intra-assembly connection probability and synaptic weights with values in the experimentally observed ranges [27,28]; and second, clustered connectivity of neurons [29]. Hence, although the concept of an excitation chain as such is a rather generic and abstract model, its basic connectivity features are consistent with experimentally observed properties. In the next section, we describe our model and its dynamics in detail. We also illustrate how activity propagation can be made faster or slower by changing synaptic weights. Then we analyze the behavior of the model and explain the role of its elements in forming the dynamics. Finally, we extend the excitation chain to a two-dimensional model, which we may call an excitation grid. We argue that this grid of assemblies can be considered as the skeleton of barrel cortex, which can generate different spatio-temporal modes of activity propagation observed experimentally in barrel cortex [30,31].
  In order to propagate a signal of activation through several excitable assemblies, we first connect several groups of neurons in a bidirectional chain (Fig 1A). Each group contains an excitatory assembly and a population of inhibitory neurons. Assemblies are defined as small populations of neurons with high connection probability. More precisely, inside each excitatory assembly synapses are strong and the connectivity is high (connection probability p = 50%) whereas each inhibitory population has smaller synaptic weights and lower connectivity. Within each group, the excitatory assembly and inhibitory population are mutually connected. Moreover, the excitatory assembly is connected to the inhibitory population and excitatory assembly of neighboring groups. In contrast, the inhibitory populations do not have inter-group connections (see Materials and methods for the details and parameters). While the previous section focused on a one-dimensional structure, in this section we consider an excitation wave in a two-dimensional grid structure inspired by the layout of barrel cortex. Barrel cortex processes sensory information from the whiskers and is a part of the rodent somatosensory system. It consists of vertical modules called barrel columns, each of which relates to one principle whisker [41]. Here we make a multicolumn model of barrel cortex which contains 25 columns, organized in the shape of 5 arcs while each arc contains 5 rows. The actual mouse barrel cortex includes 33 columns with an arc of 4 rows, 4 arcs of 5 rows and 3 arcs of 3 rows [41]. For simplicity, we only consider one cortical layer of the barrel cortex. In our model, every column consists of excitatory and inhibitory neurons. Excitatory neurons are divided into two groups, a minority of assembly neurons and a majority of non-assembly neurons (see Table 2). Fig 5 shows the schematic of the model. While assembly neurons have high internal synaptic weights and connection probability, the connections between assembly and non-assembly neurons as well as connections inside non-assembly neurons are sparse and weak. Inside a column, all three groups have connections to each other, but inter-column connections are different from intra-column connections. (see Table 2). Columns are identical in terms of number of neurons, neural parameters and connections between neurons. Inhibitory neurons in our model form only intra-column connections and are not connected to the neurons of other columns, consistent with the common assumption that inhibitory neurons send short axons and contact only local targets [42]. In contrast, excitatory neurons of each column of our model connect to the excitatory neurons of the four nearest-neighbor columns. However, due to the relatively long distance between two neighboring columns, the connection probability between columns is taken as low (p = 10%), consistent with experimental observations [43]. Just as in the excitation chain of the previous section, the dynamics of each assembly can be explained by the three-fixed-point configuration. However, the subpopulations of non-assembly neurons have low network feedback coefficients (Eq 9) so that their dynamics exhibit only the low-activity fixed point. Hence the assemblies govern the dynamics of the grid model while non-assembly neurons follow the dynamics at a lower firing rate. Neglecting non-assembly and inhibitory neurons for a moment, we can consider this barrel cortex model merely as a grid of assembly neurons. We may think of this grid as the skeleton of our model for barrel cortex. Just as in the chain model, the synaptic weight between assemblies determines the propagation speed. Here, we adjusted the value of the inter-column synaptic weight (wexc = 0. 32mV) such that the speed of activity propagation in the model is similar to that observed in the experimental data [30,31,44]. We did not need to use short-term facilitation or depression to achieve the desired speed. The remaining parameters of the model are reported in section Materials and Methods.
  We have shown that a network of spiking neurons consisting of a chain of 11 assemblies can generate reliable temporally structured activity, that extends around one second, similar to excitable media on discrete structures [50]. While two well-known models of activity propagation, the synfire chain and rate propagation, require a feedforward network structure, our excitation chain works with a bidirectional activity pattern, similar to excitable media [22–24,50]. This is advantageous because there is no direct experimental confirmation for the existence of systematic feedforward connectivity patterns in the brain [1–3,7]. Moreover, the biophysical prerequisites of the excitation chain model, namely neuronal clustering, spike-frequency adaptation, recurrent connectivity and short-range inhibitory connections, are in principle consistent with experimental findings [29,32,38,42]. Recent experimental evidence suggests a temporal evolution of activity over 500–1500ms [51,52]. The excitation chain carries the signal by a wave of synchronous activity. Seen from this perspective, it is more similar to the synfire chain than to the fluctuating rate propagation which relies on asynchronous activity and rate coding [6]. However, the excitation chain is fundamentally different from the synfire chain and rate propagation in its ability to adjust the propagation speed without modifying the synaptic delay or synaptic time constant. While the propagation speed is nearly constant in a synfire chain [53], and is very close to the synaptic time scale in fluctuating rate propagation [7] and two-dimensional systems of neurons [54], we can slow down and adjust the speed in the excitation chain by changing the strength of inter-assembly synapses. Hence, we suggest that the excitation chain can be used for population coding across slow time scales [55,56]. It is also possible that changes in synaptic weights are caused by neuro-modulators or various forms of activity-dependent synaptic plasticity. Suppose that a part of a neuronal structure which produces a complex cognitive behavior is an excitation chain, such that each assembly is responsible for performing one primitive of the behavior. Repeating the behavior, the assemblies become active one after each other in a systematic and reliable manner. In this case, a Hebbian learning rule will strengthen the synapses between excitatory assemblies and thus will work towards an increase of the execution speed of the behavior as we observed in Figs 1 and 2. This may explain why practicing sequential or rhythmic movements, such as in playing a musical instrument, can increase the speed at which the movement can be performed. The bistability of neuronal assemblies is a key component of the excitation chain model. While the mechanism of assembly activation (jumping to the high point) is based on network feedback and is independent from the neuron model, the return of the assembly to the low point (in the dormant mode) needs an element of fatigue. In our model, we use spike-frequency adaptation for this purpose. However, replacing adaption by short-term depression [35] of intra-assembly synapses yields similar results. In a previous abstract [57], we removed spike-frequency adaptation and built a chain using leaky integrate-and-fire neurons while the synapses within each assembly expressed short-term depression. With strong connections inside assemblies, and relatively weak connections between them, the chain could propagate the activation both forward and backward. Similar to what is presented here, the propagation speed was tunable by changing the inter-assembly maximum synaptic weights. In a related model, a feedforward chain of neural assemblies with strong recurrent synapses which express short-term depression will propagate activity with propagation latency over 10 assemblies between 90ms and 200ms [9]. In contrast to this earlier model, we use symmetric connectivity so that propagation has no preferred direction. Another related work [58] studied the influence of short-term synaptic dynamics on switching activity patterns in networks that contain attractors. One may wonder about the plausibility of neural assemblies in cortical networks. Applying synaptic plasticity rules in a randomly connected network in a balanced state [59,60] leads to subgroups of neuron with strong recurrent connections [61]. A recent study used spike-timing dependent plasticity rules acting on excitatory to excitatory [62] and inhibitory to excitatory connections [63] as well as a synaptic scaling rule [64]. It was found that neurons which initially fire at higher firing rates compared to other neurons form subgroups with strong inward synapses and larger mean of outward synaptic weights after applying plasticity rules [61]. One way of extending our work is embedding the excitation chain in a bigger background network [5,16,17]. Then, excitatory assemblies and inhibitory populations receive synaptic input from the neurons of the larger background network. In case of a balanced background network, we would expect that the chain will exhibit results similar to those shown here, except that due to variance of synaptic input, the activation times will show more jitter compared to the result we obtained (Figs 1D, 1E and 2C). Constructing a balanced network for adapting neurons will be possible using recent mean field methods [65] and is left for future work. It would also be interesting to further explore bidirectional synaptic connections between excitatory and additional inhibitory neurons within the same group. Our expectation is that, if parameters are tuned to get a balanced state, the peak firing rate of neurons inside the assembly would be significantly reduced without a substantial change in the speed of signal propagation across the chain; cf. Fig 4. In the grid model, assemblies are responsible for transmitting the activity between barrel columns. Removing the inhibitory and non-assembly neurons from the model, the grid of assemblies is able to propagate and circulate the activity on its own. Therefore we consider the grid of assemblies as the skeleton of the model, while the other neurons just follow the activity wave. This, however, is not meant to imply that inhibitory and non-assembly neurons do not play an important role in the cortex. Whereas in our model, we only consider the anesthetized state, information processing in the awake cortex very likely involves more than just the assembly neurons and requires the contribution of the other neurons. In the literature, there exist several competition-based models for reproducing cortical trial-to-trial variability [66–69] and for modeling working memory using continuous attractors [70,71]. In these models, individual neurons or neural assemblies (also called neural clusters) try to become active and to suppress others by direct inhibitory connections or global inhibition. However, in the two models considered here (the chain and grid of assemblies) the situation is completely different. Instead of competition, the assemblies cooperate with each other to propagate activity signals through the tissue. Hence, in this type of model there is no need for global inhibition. Similar to bump attractor models proposed for implementing working memory [70,71] and attractor maps used for encoding spatial location in hippocampus [72–74], our model relies on strong recurrent excitatory connections. However, while these encoding-models are designed to stabilize neuronal activity in time in order to provide a stationary memory of a continuous variable, our model’s aim is to propagate the activity with a specified speed and path to relay sensory information or to perform slow behavioral tasks. Activity propagation was studied previously using the Wilson-Cowan model of neuronal population dynamics [22,23] and the dynamics of neural fields [24,25,50,75,76]. However, it is often hard to link network parameters used in these abstract models to neuronal and synaptic parameters. In our approach, we start directly from neural parameters extracted from experiments [38,39] and explore the influence of synaptic weight or short-term plasticity [35]. Despite the richness of the biological parameter space, the essential mathematical features of excitable media [22–25] are still apparent in the models examined here. In the literature, there exist several studies [77,78] that observed traveling activity waves in a balanced network with distance dependent connectivity structure. The multicolumn model presented in this work is able to produce dynamics similar to the spontaneous regime of cortical activity [31]. However, it is different from resting-state networks [79–83] designed for explaining the synchrony of different brain regions. In these models, each brain region is typically considered as a nonlinear oscillator. The coupling strength between oscillators [79] as well as synaptic transmission delay and noise [80,81] are tuned such that different brain regions show synchrony similar to observed data. While such models are very successful in representing resting-state activity, their focus on macroscopic network structure may limit the range of dynamics produced by the models. In contrast, in our model we focus on a particular element of neuronal circuit-level connectivity, namely local assemblies of neurons, which may be contained in a small part of each cortical column. While such assemblies may make only a small contribution to the average activity, macroscopic propagation of activity across barrel columns or brain regions may depend on such assemblies. In systems with excitable elements such as ours, the type of dynamics strongly depends on initial conditions of the neurons and nonlinearly on the inputs [84]. Therefore embedded excitable assemblies provide nolinear mesoscopic processing characteristics to the circuits that may easily be overlooked in resting-state or other macroscopic models relying on averaged measures of connectivity between areas. To conclude we would like to highlight the predictive aspect of our work. So far, neural assemblies have been used for explaining working memory [70,71], cortical trial-to-trial variability [66–69] and slow oscillations in the cortex [34]. We suggest that beside these roles, neural assemblies are also responsible for activity with adjustable speed in the cortex [52]. Consider a complex behavioral task which includes a sequence of several subtasks. If activation of each assembly of the chain initiates a subtask, the whole chain is able to perform the complex tasks with desirable speed. In case of repeating the tasks, Hebbian learning strengthens interassembly synapses and faster propagation. Therefore, the cortical network is able to perform the task faster after practicing the task. Because of symmetric connectivity, we could explain bidirectional activity propagation observed in the experiments [32]. Furthermore, our work predicts that inhibitory neurons reduce the propagation speed. In a multi-array recording [32] with GABA receptor antagonists near the electrodes of array, we therefore predict an increase in the propagation speed compared to the control condition.
  Neuronal parameters used in the simulations are reported in Table 1. Table 2 summarizes the network parameters. As our neuron model we use a current-based generalized integrate-and-fire (GIF) model which implements spike-frequency adaptation using a spike-triggered current and a moving firing-threshold mechanisms [38]. The dynamics of the neuron’s sub-threshold membrane potential (V (t) ) is described by: C d V (t) d t = - g L (V (t) - E L) - ∑ t j ^ < t η (t - t j ^) + I (t) (1) where parameters C, gL and EL are the passive parameters of the neuron. I (t) is the synaptic input and η (t) is the shape of the spike-triggered current caused by spikes of the neuron itself at times t ^ j. After each spike emission, the membrane potential is reset to Vreset, integration of Eq 1 restarts and the neuron goes through an absolute refractory period of duration τref. Spikes are produced stochastically (similar to an inhomogeneous Poisson process) with the firing intensity: λ (t) = λ 0 exp (V (t) - V T (t) Δ V) (2) where λ0 is the stochastic intensity at the firing threshold VT, and ΔV is a constant which defines the level of stochasticity. The threshold VT follows the dynamic: V T (t) = V T * + ∑ t j ^<t γ (t - t j ^) (3) where V T * is constant and γ (t) describes the time course of the threshold after a spike emission. In Eq (1), the synaptic input Ii (t) received by neuron i is determined by the spikes of synaptically connected neurons: I i (t) = ∑ j w i j ∑ f α (t - t j f) = ∑ j w i j ∫ 0 ∞ α (s) S j (t - s) d s (4) where wij is the weight of the synapse connecting neuron j to neuron i, and α (t) = e - (t - Δ) / τ syn for t ≥ Δ is the post-synaptic current (PSC) shape. The synaptic transmission delay (Δ) in all simulation is 1ms. S j (t) = ∑ k δ (t - t j k) is the spike train of neuron j, δ denotes the Dirac δ-function and t j k is the kth spike of neuron j. The synaptic weight wij indicates the PSC amplitude. Given the neuronal parameters, one can relate the PSC to the post-synaptic potential (PSP). We report values both of PSC and PSP amplitudes used in our simulations. We ran simulations using the Brian simulator [85] for simulating the chain and NEST [86] for simulating the grid. In order to initiate the activity in the chain, we stimulate one assembly of the chain (for example the first assembly, the last assembly or an assembly in the middle of the chain). For stimulating an assembly, we connect each of its neurons to 25 Poisson neurons which fire with the rate of 5Hz each for 25ms. The synaptic weight between the Poisson neurons and the assembly neurons is 0. 18nA. Here we present some theory necessary to explain the dynamics of excitable assemblies. The dynamics of a neuronal population can be described by two equations relating the firing rate averaged over all neurons and the mean of the synaptic input received by them. The first relation is called the neuron’s gain function. Injecting a weakly fluctuating current Isyn into a neuron produces an average firing rate of r = g (〈 I syn 〉, σ I) (5) where 〈Isyn〉 and σI are the average and the standard deviation of the synaptic current over time, respectively, and g is the gain function. Although there are ways to compute the firing rate of adaptive integrate-and-fire neuron models in closed-form [87,88] or by using a self-consistent numerical approach [89–91], there is no straightforward analytical solution for computing the gain function of the GIF model that we use here. We obtain the gain function (5) by numerical simulation [34]. For the simulations to determine the gain function numerically the injected current is given by I (t) = 〈 I syn 〉 + σ I q 2 ∫ 0 ∞ α (s) ξ (t - s) d s (6) where ξ (t) is white noise with mean 〈ξ (t) 〉 = 0 and covariance 〈ξ (t) ξ (t′) 〉 = δ (t − t′), α (t) is the shape of the PSC defined above and q 2 = ∫ 0 ∞ α 2 (t) d t. Depending on the duration of injection, the neuron goes into different adaptation states. By injecting the current (6) for a short episode of 10ms, we can estimate the firing rate in the non-adapted state. In case of a longer stimulation period, we can divide the time into intervals of 10ms and extract the rate-current relation in the different, progressively more adapted states. This method has been used to obtain the gain functions displayed in Fig 3A and 3B. The network activity gives rise to the second relation between the average firing rate and the average synaptic current. The synaptic input of an arbitrary neuron i is described by: I i, syn = ∑ j w i j (∫ 0 ∞ α (s) S j (t - s) d s) (7) where wij is the weight of the synapse connecting neuron j to neuron i and Sj (t) is the spike train of neuron j. The sum runs over all other neurons j in the assembly. Averaging both sides over time and input neurons gives the average input current: 〈Isyn〉 = Npqwr, where N is the number of neurons inside the population, p is the connection probability between neurons, w is the synaptic weight and q is the total charge of one PSC pulse: q = ∫ 0 ∞ α (t) d t. Rearranging this equation yields: r = 〈 I syn 〉 N p q w (8) We refer to the denominator of Eq 8 as the network feedback coefficient (Cfb) of the population [34]: C fb = N p q w (9) We use these two relations for analysis of the behavior of excitatory assemblies in the “Results” section. Our aim is to estimate the difference between activation times of two consecutive assemblies (which is a measure of propagation speed) using an analytical approach. We assume that the values of the parameters are given and the time course r (t) of the population rate signal of an arbitrary assembly in the chain is known. Furthermore, we assume that the populations are silent before activation (r (t) = 0 for t < ta). Suppose that we have two excitatory assemblies, assembly 1 and assembly 2. We refer to their activation time as t ¯ 1 and t ¯ 2 respectively. The aim of the calculation is to find the difference in activation times x = t ¯ 2 - t ¯ 1. Recall that the activation time has been defined above as the expected time at which all assembly neurons have spiked once. Assuming independent Poisson firing of the assembly with rate r (t), we can find t ¯ 1 as the moment when the expected spike count reaches 1, n (0; t ¯ 1) / N = 1, where n (a; b) is the number of spikes in time interval [a, b) and N is the number of neuron in the assembly. Inserting the shape of assembly population rate we obtain ∫ 0 t ¯ 1 r (t) d t = 1 (10) which we can solve for t ¯ 1. Note, however, that the Poisson assumption made here does not ensure that no neuron fires more than one spike before t ¯ 1. Therefore Eq 10 yields an approximate value of t ¯ 1. Finding the value of t ¯ 2 is more complicated. After activation of assembly 1, its neurons send synaptic input to assembly2. The average input received by each assembly2 neuron can be computed: I syn_fwd (t) = N p exc w exc (α (t) * r (t - Δ) ) (11) where pexc and wexc are inter-assemblies connection probability and synaptic weight respectively, * denotes the convolution operator and Δ is synaptic transmission delay. However, this is not the only synaptic input received by neurons in assembly2. Even before t ¯ 2, several neurons of assembly2 may already fire spikes (due to random fluctuations) and send some feedback current to other neurons. This averaged current can be computed similarly: I syn_self (t) = (N - 1) p self w self (α (t) * r (t - Δ - x) ) (12) where pself and wself are intra-assemblies connection probability and synaptic weight respectively, and x is the difference of activation times. Note that the shape of r (t) is assumed to be the same for all assemblies of the chain. However, we must be careful about the timing of each current. Suppose that at the time t = 0 assembly 1 starts to fire, therefore assembly2 receives a first input at the time t = Δ. After a while, assembly2 starts to fire. The difference between these two starting times is denoted by x. Therefore the feedback current is received by assembly2 neurons at the time t = Δ + x. The total synaptic input received by neurons of assembly2 is the summation of Isyn_fwd and Isyn_self. Consequently, we can write the total input received by assembly2 neurons as I syn (t) = I syn_fwd (t) + I syn_self (t) (13) Now we can calculate the subthreshold membrane potential of neurons in assembly2 by solving Eq 1: V (t) = E L + 1 C (e - t / τ m * I syn (t) ) (14) Note that since we want to calculate the time of first spikes of neurons, we can neglect the spike-triggered current and the moving firing threshold. The firing intensity is then given as a function of V (t) by Eq (2), except that V T (t) = V T * because we assumed that neurons are not adapted and all γ (t) equal zero. Then using the distribution of first spikes P (t), we are able to calculate the average time of first spikes of assembly2: P (t) = λ (t) exp (- ∫ 0 t λ (t ′) d t ′) (15) t ¯ 2 = ∫ 0 ∞ t P (t) d t (16) Finally, using Eqs 10 and 16, we can calculate the time difference between activation times: x = t ¯ 2 - t ¯ 1 (17) Note, however, that t ¯ 2 in Eq (17) depends on the value of x through Eq (12). Therefore, we have formed a self-consistent equation for x. We feed this value in Eq 12 and get it back in Eq 17. If the output value of x equals its input value, we found the proper value. Using a simple search, we are able to find this value numerically. We apply this approach for our chain and calculate the value of x for different values of wexc; these results are presented in Fig 3D. Note that we need to obtain the shape of the assembly population rate r (t) by neural simulation beforehand. However, the same shape r (t) can be used for all values of wexc, because for pexcwexc ≪ pselfwself the time course of the initial rise is dominated by the self-feedback (see Fig 3B). We use short-term plasticity [35,36] in one series of our simulations (Fig 1F). This synaptic model supposes that each synapse has a certain amount of resources (such as neurotransmitter packed in vesicles ready for release) denoted by x, with dynamics d x d t = 1 - x τ rec - uxδ (t - t f) (18) d u d t = U - u τ facil + U (1 - u) δ (t - t f) (19) where U (jump of release fraction), τrec (recovery time constant) and τfacil (facilitation time constant) are three parameters of the model. u (release fraction) and x are the two variables of the system. Whenever a neuron fires a spike (tf denotes the firing time), it produces a PSC with amplitude of uxw (w is the synaptic weight). Then, the amount of resource is decreased by ux and the release fraction is increased by U (1 − u). In our simulations, we either used facilitation or depression. We chose the values τrec = 0. 001ms and τfacil = 500ms for the facilitation case and τrec = 800ms and τfacil = 0. 001ms for the depression case. The value of U is different in each simulation. Note that for the depression case, we have to fix the amplitude of the first PSC regardless of value of U. We did that by adjusting the value of the synaptic weight. We simulate different trials of the grid with shuffled initial values (Fig 7). The aim of shuffling is to show that many initial patterns will lead to activity circulation and not just the tuned pattern of Fig 6D (right). For the shuffling, we take mean values of γ2 (0) (initial value of the moving threshold kernel with longer time constant) of each column in Fig 6D (right) and randomly assign these means to columns. Then, using these means, we randomly choose the value of γ2 (0) for each neuron in the columns.
summary: Models of activity propagation in cortical networks have often been based on feedforward structures. Here we propose a model of activity propagation, called excitation chain, which does not need such a feedforward structure. The model is composed of excitable neural assemblies with spike-frequency adaptation, connected bidirectionally in a row or a grid. This prototypical neural circuit can propagate activity forwards, backwards or in both directions. Furthermore, the propagation speed is slow enough to trigger the generation of behaviors on the time scale of hundreds of milliseconds. A two-dimensional variant of the model is able to generate different activity propagation patterns, similar to spontaneous activity and stimulus-evoked responses in anesthetized mouse barrel cortex. We propose the excitation chain model as a basic component that can be employed in various ways to create spiking neural circuit models that generate signals on behavioral time scales. In contrast to abstract models of excitable media, our model makes an explicit link to the time scale of neuronal spikes.
keywords: action potentials
  medicine and health sciences
  neural networks
  nervous system
  membrane potential
  electrophysiology
  neuroscience
  simulation and modeling
  synaptic plasticity
  neuronal plasticity
  research and analysis methods
  computer and information sciences
  developmental neuroscience
  animal cells
  neural pathways
  cellular neuroscience
  neuroanatomy
  cell biology
  anatomy
  synapses
  physiology
  neurons
  biology and life sciences
  cellular types
  neurophysiology
title: Excitable neuronal assemblies with adaptation as a building block of brain circuits for velocity-controlled signal propagation