- Email: [email protected]
Self-oscillatory dynamics in recurrent excitatory networks
Neurocomputing 44–46 (2002) 257 – 262
www.elsevier.com/locate/neucom
Self-oscillatory dynamics in recurrent excitatory networks Fabi&an P. Alvareza;∗;1 , Jean-Fran-cois Vibertb a Facultad
de Ciencias, Igua 4225 piso 4 11400, Montevideo, Uruguay U444, CHU Saint-Antoine, Paris, France
b INSERM
Abstract We present results concerning the capacity of recurrent excitatory neural networks to reach self-sustained oscillatory 1ring. We simulated networks made up of leaky integrate-and-1re based neurons, with excitatory connections. We studied the conditions of emergence of sustained self-oscillations, in which synchronic and asynchronic activities alternate, leading to a network bursting pattern. Finally, we analyzed the encoding by the network of external inputs depending on its 1ring activity. The results support the idea that the network encoding may c 2002 Elsevier Science B.V. All rights reserved. change along time. Keywords: Recurrent excitatory network; Self-oscillation; Adaptation
1. Introduction The central nervous system (CNS) encodes and integrates the information which arrives from external world through di8erent sensors and pathways. At a macroscopic view, the CNS displays a wide range of spatially correlated rhythmic patterns. The correlation in the 1ring times of neurons might encode such external information. There is increasing evidence for neural coding through precise spiking timing of neurons, but the way by which this encoded information propagates through active networks is still an open problem [3]. Several works have shown the importance of synaptic inhibitions in network synchrony [2,10,11,15, among others]. Nevertheless, some of the structures of the CNS ∗
Corresponding author. E-mail addresses: [email protected] (F.P. Alvarez), [email protected] (J.-F. Vibert). 1 Present address: UNIC, CNRS UPR 2191, 91198 Gif-sur-Yvette, France. Partially supported by CSIC-UdelaR. c 2002 Elsevier Science B.V. All rights reserved. 0925-2312/02/$ - see front matter PII: S 0 9 2 5 - 2 3 1 2 ( 0 2 ) 0 0 3 4 5 - 4
258
F.P. Alvarez, J.-F. Vibert / Neurocomputing 44–46 (2002) 257 – 262
involve purely excitatory connections, leading to recurrent excitatory networks (RENs) [1,5]. In RENs, the action potentials reverberate through excitatory connections, and then, the neurons receive back their own outputs after a delay. Simulated RENs exhibit di8erent 1ring regimes according to the interneuronal delay [14,13]. RENs constitute positive feedback systems, that should result in stationary saturated 1ring. Nevertheless, RENs avoid the saturated 1ring, due to the adaptation present in biological neurons [14]. Moreover, these neurons receive inputs from multiple sources in the nervous system. Those, and the reentrant inputs arriving after variable delays, shape a large amount of uncorrelated inputs assembled as noise. The purpose of the present work is to complement previous investigations concerning the dynamics of simulated RENs [7,13], with the study of the emergence of self-sustained rhythmic patterns in networks of neurons which present adaptation to the repetitive 1ring. Finally, we show how the 1ring state of the network may a8ect the coding of external inputs. 2. Methods The neurons were modeled by a modi1ed leaky integrate and 1re model (MLIM). The MLIM adds two features to the classical leaky integrator: the possibility of setting the after-spike reset voltage at a value di8erent from the resting potential; and a fatigue term, accounting for neuron adaptation. This adaptation was modeled as a post-spike elevation of threshold, that decays exponentially to its rest value [9]. We considered networks made of N neurons (100 ¡ N ¡ 500), with each neuron connected randomly to K neurons (5 ¡ K ¡ N ). We refer to as connectivity the value of K. Each EPSP was considered as an instantaneous jump in the membrane potential (of amplitude VEPSP ), which then decays exponentially. The noise impinging upon the neurons was modeled by additive Gaussian white noise, with zero mean and standard deviation . The dynamics of the network was characterized by its global activity, namely the percentage of spiking neurons at a given instant. The simulations were made with the XNBC neural network simulation toolkit [12] (http://www.b3e.jussieu.fr/xnbc). 3. Results The self-oscillatory dynamics (SOD) concerns the global activity of the network. Simulations started with random values of the membrane potential, and then, noisy input was given to each neuron. Because of the excitatory connections, the network was driven to a highly synchronic 1ring regime (phasic network activity). In this regime, a high percentage (which depended on parameters values) of neurons spiked simultaneously for several consecutive 1rings. Nevertheless, noise and refractory periods contributed to desynchronize the network 1ring, leading the global activity to collapse down to low percentages (asynchronic tonic activity). The network remained at this state until, suddenly, the synchronized behavior arose again, and the cycle went on. Therefore, without any change in external conditions, the network activity oscillated,
F.P. Alvarez, J.-F. Vibert / Neurocomputing 44–46 (2002) 257 – 262 (A)
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Fig. 1. Self-oscillatory dynamics. Raster plot (upper panels) and network activity plot (lower panels) for a network of 200 neurons and K = 40. A. Several cycles along time; B. short temporal window to display the emergence of synchronization.
driven by the Luctuations in the membrane potential, with alternating synchronic and asynchronic activities leading to a network bursting pattern. Fig. 1 shows an example of this pattern, for a 200 neurons network (only 100 are plotted in the raster plots). Fig. 1A represents several cycles of the SOD, whereas Fig. 1B shows a detailed view of the emergence of the synchronization. Note that, even if the peak of the activity is about 40 – 45% of the neurons, almost all the neurons 1re at each “network 1re”, but slightly desynchronized. A higher synchronization may be reached for other set of parameters (e.g. higher network connectivity, see below). To characterize this 1ring regime, we simulated networks between 100 and 500 neurons, for di8erent values of resting potential, threshold, post-spike reset, threshold fatigue, time constants. The SOD arose for a limited range of the parameters. Intra-network connectivities from 10% to 100% were explored. The SOD appeared as long as the synaptic weights are changed concomitantly, which indicates that the relevant variable is the synaptic weight=connection ratio [7]. Di8erent ranges of interneuronal delays were tested. For short delays, the network reached the SOD, while if delays were increased, the network was synchronized by the neurons’ bursting [14,13], and the asynchronic activity could not been reached again. Fig. 2 shows the e8ect of one of the parameters analyzed (the connectivity K) upon the network’s dynamics. For increasing connectivities, two modi1cations of the dynamics were apparent: the maximum activity level during the phasic activity increased, and the extent of the asynchronic tonic activity diminished. Note that for the classic LIM (1xed threshold), the network get stuck in a fully synchronized regime for connectivities higher than K = 9 [7] (for parameter values as the ones presented here). Nevertheless, for the present model, the SOD emerged even for the full connected
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Fig. 2. E8ect of the connectivity on the network activity. All panels: N =200. Upper left: K =20, upper right: K = 60, bottom left: K = 120, bottom right: K = 200 (full connectivity). Abscissae: time in ms. Ordinates: network activity (in number of 1ring neurons).
network (K = 200, lower right panel in Fig. 2), which indicates that the adaptation plays a major role in the network dynamics. As in the example presented in Fig. 1, each neuron considered individually is 1ring all the time, the oscillation occurred in the coherence of the network 1ring. Di8erent sets of noisy inputs were studied. If the non-noisy input was strong enough to produce a pacemaker behavior in a single isolated MLIM, the SOD emerged when noise was added. This was also the case for slightly lower inputs, that drove the membrane potential close to the threshold in the absence of noise. This last case was the initial condition used for all the simulations presented in this work. When input was further decreased, the SOD disappeared. This suggests that the SOD is related to the “refractory regime” [8] of the MLIM. As we did for connectivity, we explored the network dynamics for di8erent values of the noise standard deviation. The SOD appears whereas noise was held at moderate values ( ¡ 3), and, as it could be expected, increasing values of facilitates the asynchronic tonic regime (not shown here). Finally, we analyzed the network answer to external inputs, as a 1rst approach to characterize how this oscillating network may encode inputs, depending on its 1ring state. Fig. 3 shows examples of such encoding. All the neurons received simultaneously the same external input (an EPSP of amplitude VEPSP = 5 mV), at the instants pointed by the arrows (chosen during the simulation). In the left panel (Fig. 3A), the example shows the case where the input did not a8ect the dynamics, in the sense that the ongoing 1ring was the one we should expect if the input was not present. The right panel (Fig. 3B) shows the opposite case, in which the phasic network 1ring was suppressed by the external stimulus (t =3000 ms). The di8erent responses of the network according
F.P. Alvarez, J.-F. Vibert / Neurocomputing 44–46 (2002) 257 – 262
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Fig. 3. E8ect of external input on the network activity. Arrows indicate the instants at which an external input (5 mV instant depolarization) is added to all the neurons of the network. Abscissae: time in ms. Ordinates: network activity (in number of 1ring neurons).
to its 1ring state, would deserve a further characterization to get deeper understanding of this system. 4. Discussion According to these results, even in the presence of a sustained external input, the activity of a purely excitatory network may oscillate along time. Two components are essential to get the oscillatory dynamics for this system: noisy inputs and adaptation of neurons to repetitive 1ring (fatigue). This oscillatory regime implies the switching between two states: a synchronic (phasic) network 1ring, and an asynchronic (tonic) one. The alternance between both states is self-sustained. Neuronal coding has been a common source of debates during last decades. A wide range of coding schemes has been proposed, with many experimental systems accounting for them [4]. We studied here a network of MLIMs which may switch between two behaviors, and then capable of switch between di8erent coding of the a8erent inputs. This self-oscillating behavior supports the idea that the coding is not a static problem, but a dynamic one. These results are challenging since increasing importance have been assigned to bursting in neural processing [6], and hence, this simple system provides a model for studying its incidence in network dynamics. References [1] M. Abeles, Corticonics: Neural Circuits of the Cerebral Cortex, Cambridge University Press, Cambridge, 1991. [2] N. Brunel, Dynamics of networks of randomly connected excitatory and inhibitory spiking neurons, J. Physiol. (Paris) 94 (2000) 445–463. [3] M. Diesmann, M.-O. Gewaltig, A. Aertsen, Stable propagation of synchronous spiking in cortical neural networks, Nature 402 (1999) 529–533. [4] J.J. Eggermont, Is there a neural code? Neurosci. Biobehav. Rev. 22 (1998) 355–370. [5] G. Fortin, J. Champagnat, Spontaneous synaptic activities in rat nucleus tractus solitarius neurons in vitro, Brain Res. 630 (1993) 125–135.
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[6] L. Menendez de la Prida, N. Stollenwerk, J.V. Sanchez-Andres, Bursting as a source for predictability in biological neural network activity, Phys. D 110 (1997) 323–331. [7] J. Pham, K. Pakdaman, J. Champagnat, J.-F. Vibert, Activity in sparsely connected excitatory neural networks: e8ect of connectivity, Neural Networks 11 (1998) 415–434. [8] T.W. Troyer, K.D. Miller, Physiological gain leads to high ISI variability in a simple model of a cortical regular spiking cell, Neural Comput. 9 (1997) 971–983. [9] H.C. Tuckwell, Introduction to Theoretical Neurobiology, Cambridge University Press, Cambridge, 1988. [10] C. van Vreeswijk, L.F. Abbott, G.B. Ermentrout, When inhibition not excitation synchronizes neural 1ring, J. Comput. Neurosci. 1 (1994) 313–321. [11] C. van Vreeswijk, D. Hansel, Patterns of synchrony in neural networks with spike adaptation, Neural Comput. 13 (2001) 959–992. [12] J.-F. Vibert, F. Alvarez, E. Kosmidis, XNBC V9: A user friendly simulation and analysis tool for neurobiologists, Neurocomputing 38– 40 (2001) 1715–1723. [13] J.-F. Vibert, F. Alvarez, J. Pham, E8ect of transmission delays and noise in recurrent excitatory neural networks, BioSystems 48 (1998) 255–262. [14] J.-F. Vibert, K. Pakdaman, N. Azmy, Interneural delay modi1cation synchronizes biologically plausible neural networks, Neural Networks 7 (1994) 589–607. [15] J.A. White, Synchronization and oscillatory dynamics in heterogeneous, mutually inhibited neurons, J. Comput. Neurosci. 5 (1998) 5–16.
www.elsevier.com/locate/neucom
Self-oscillatory dynamics in recurrent excitatory networks Fabi&an P. Alvareza;∗;1 , Jean-Fran-cois Vibertb a Facultad
de Ciencias, Igua 4225 piso 4 11400, Montevideo, Uruguay U444, CHU Saint-Antoine, Paris, France
b INSERM
Abstract We present results concerning the capacity of recurrent excitatory neural networks to reach self-sustained oscillatory 1ring. We simulated networks made up of leaky integrate-and-1re based neurons, with excitatory connections. We studied the conditions of emergence of sustained self-oscillations, in which synchronic and asynchronic activities alternate, leading to a network bursting pattern. Finally, we analyzed the encoding by the network of external inputs depending on its 1ring activity. The results support the idea that the network encoding may c 2002 Elsevier Science B.V. All rights reserved. change along time. Keywords: Recurrent excitatory network; Self-oscillation; Adaptation
1. Introduction The central nervous system (CNS) encodes and integrates the information which arrives from external world through di8erent sensors and pathways. At a macroscopic view, the CNS displays a wide range of spatially correlated rhythmic patterns. The correlation in the 1ring times of neurons might encode such external information. There is increasing evidence for neural coding through precise spiking timing of neurons, but the way by which this encoded information propagates through active networks is still an open problem [3]. Several works have shown the importance of synaptic inhibitions in network synchrony [2,10,11,15, among others]. Nevertheless, some of the structures of the CNS ∗
Corresponding author. E-mail addresses: [email protected] (F.P. Alvarez), [email protected] (J.-F. Vibert). 1 Present address: UNIC, CNRS UPR 2191, 91198 Gif-sur-Yvette, France. Partially supported by CSIC-UdelaR. c 2002 Elsevier Science B.V. All rights reserved. 0925-2312/02/$ - see front matter PII: S 0 9 2 5 - 2 3 1 2 ( 0 2 ) 0 0 3 4 5 - 4
258
F.P. Alvarez, J.-F. Vibert / Neurocomputing 44–46 (2002) 257 – 262
involve purely excitatory connections, leading to recurrent excitatory networks (RENs) [1,5]. In RENs, the action potentials reverberate through excitatory connections, and then, the neurons receive back their own outputs after a delay. Simulated RENs exhibit di8erent 1ring regimes according to the interneuronal delay [14,13]. RENs constitute positive feedback systems, that should result in stationary saturated 1ring. Nevertheless, RENs avoid the saturated 1ring, due to the adaptation present in biological neurons [14]. Moreover, these neurons receive inputs from multiple sources in the nervous system. Those, and the reentrant inputs arriving after variable delays, shape a large amount of uncorrelated inputs assembled as noise. The purpose of the present work is to complement previous investigations concerning the dynamics of simulated RENs [7,13], with the study of the emergence of self-sustained rhythmic patterns in networks of neurons which present adaptation to the repetitive 1ring. Finally, we show how the 1ring state of the network may a8ect the coding of external inputs. 2. Methods The neurons were modeled by a modi1ed leaky integrate and 1re model (MLIM). The MLIM adds two features to the classical leaky integrator: the possibility of setting the after-spike reset voltage at a value di8erent from the resting potential; and a fatigue term, accounting for neuron adaptation. This adaptation was modeled as a post-spike elevation of threshold, that decays exponentially to its rest value [9]. We considered networks made of N neurons (100 ¡ N ¡ 500), with each neuron connected randomly to K neurons (5 ¡ K ¡ N ). We refer to as connectivity the value of K. Each EPSP was considered as an instantaneous jump in the membrane potential (of amplitude VEPSP ), which then decays exponentially. The noise impinging upon the neurons was modeled by additive Gaussian white noise, with zero mean and standard deviation . The dynamics of the network was characterized by its global activity, namely the percentage of spiking neurons at a given instant. The simulations were made with the XNBC neural network simulation toolkit [12] (http://www.b3e.jussieu.fr/xnbc). 3. Results The self-oscillatory dynamics (SOD) concerns the global activity of the network. Simulations started with random values of the membrane potential, and then, noisy input was given to each neuron. Because of the excitatory connections, the network was driven to a highly synchronic 1ring regime (phasic network activity). In this regime, a high percentage (which depended on parameters values) of neurons spiked simultaneously for several consecutive 1rings. Nevertheless, noise and refractory periods contributed to desynchronize the network 1ring, leading the global activity to collapse down to low percentages (asynchronic tonic activity). The network remained at this state until, suddenly, the synchronized behavior arose again, and the cycle went on. Therefore, without any change in external conditions, the network activity oscillated,
F.P. Alvarez, J.-F. Vibert / Neurocomputing 44–46 (2002) 257 – 262 (A)
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Fig. 1. Self-oscillatory dynamics. Raster plot (upper panels) and network activity plot (lower panels) for a network of 200 neurons and K = 40. A. Several cycles along time; B. short temporal window to display the emergence of synchronization.
driven by the Luctuations in the membrane potential, with alternating synchronic and asynchronic activities leading to a network bursting pattern. Fig. 1 shows an example of this pattern, for a 200 neurons network (only 100 are plotted in the raster plots). Fig. 1A represents several cycles of the SOD, whereas Fig. 1B shows a detailed view of the emergence of the synchronization. Note that, even if the peak of the activity is about 40 – 45% of the neurons, almost all the neurons 1re at each “network 1re”, but slightly desynchronized. A higher synchronization may be reached for other set of parameters (e.g. higher network connectivity, see below). To characterize this 1ring regime, we simulated networks between 100 and 500 neurons, for di8erent values of resting potential, threshold, post-spike reset, threshold fatigue, time constants. The SOD arose for a limited range of the parameters. Intra-network connectivities from 10% to 100% were explored. The SOD appeared as long as the synaptic weights are changed concomitantly, which indicates that the relevant variable is the synaptic weight=connection ratio [7]. Di8erent ranges of interneuronal delays were tested. For short delays, the network reached the SOD, while if delays were increased, the network was synchronized by the neurons’ bursting [14,13], and the asynchronic activity could not been reached again. Fig. 2 shows the e8ect of one of the parameters analyzed (the connectivity K) upon the network’s dynamics. For increasing connectivities, two modi1cations of the dynamics were apparent: the maximum activity level during the phasic activity increased, and the extent of the asynchronic tonic activity diminished. Note that for the classic LIM (1xed threshold), the network get stuck in a fully synchronized regime for connectivities higher than K = 9 [7] (for parameter values as the ones presented here). Nevertheless, for the present model, the SOD emerged even for the full connected
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F.P. Alvarez, J.-F. Vibert / Neurocomputing 44–46 (2002) 257 – 262
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Fig. 2. E8ect of the connectivity on the network activity. All panels: N =200. Upper left: K =20, upper right: K = 60, bottom left: K = 120, bottom right: K = 200 (full connectivity). Abscissae: time in ms. Ordinates: network activity (in number of 1ring neurons).
network (K = 200, lower right panel in Fig. 2), which indicates that the adaptation plays a major role in the network dynamics. As in the example presented in Fig. 1, each neuron considered individually is 1ring all the time, the oscillation occurred in the coherence of the network 1ring. Di8erent sets of noisy inputs were studied. If the non-noisy input was strong enough to produce a pacemaker behavior in a single isolated MLIM, the SOD emerged when noise was added. This was also the case for slightly lower inputs, that drove the membrane potential close to the threshold in the absence of noise. This last case was the initial condition used for all the simulations presented in this work. When input was further decreased, the SOD disappeared. This suggests that the SOD is related to the “refractory regime” [8] of the MLIM. As we did for connectivity, we explored the network dynamics for di8erent values of the noise standard deviation. The SOD appears whereas noise was held at moderate values ( ¡ 3), and, as it could be expected, increasing values of facilitates the asynchronic tonic regime (not shown here). Finally, we analyzed the network answer to external inputs, as a 1rst approach to characterize how this oscillating network may encode inputs, depending on its 1ring state. Fig. 3 shows examples of such encoding. All the neurons received simultaneously the same external input (an EPSP of amplitude VEPSP = 5 mV), at the instants pointed by the arrows (chosen during the simulation). In the left panel (Fig. 3A), the example shows the case where the input did not a8ect the dynamics, in the sense that the ongoing 1ring was the one we should expect if the input was not present. The right panel (Fig. 3B) shows the opposite case, in which the phasic network 1ring was suppressed by the external stimulus (t =3000 ms). The di8erent responses of the network according
F.P. Alvarez, J.-F. Vibert / Neurocomputing 44–46 (2002) 257 – 262
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Fig. 3. E8ect of external input on the network activity. Arrows indicate the instants at which an external input (5 mV instant depolarization) is added to all the neurons of the network. Abscissae: time in ms. Ordinates: network activity (in number of 1ring neurons).
to its 1ring state, would deserve a further characterization to get deeper understanding of this system. 4. Discussion According to these results, even in the presence of a sustained external input, the activity of a purely excitatory network may oscillate along time. Two components are essential to get the oscillatory dynamics for this system: noisy inputs and adaptation of neurons to repetitive 1ring (fatigue). This oscillatory regime implies the switching between two states: a synchronic (phasic) network 1ring, and an asynchronic (tonic) one. The alternance between both states is self-sustained. Neuronal coding has been a common source of debates during last decades. A wide range of coding schemes has been proposed, with many experimental systems accounting for them [4]. We studied here a network of MLIMs which may switch between two behaviors, and then capable of switch between di8erent coding of the a8erent inputs. This self-oscillating behavior supports the idea that the coding is not a static problem, but a dynamic one. These results are challenging since increasing importance have been assigned to bursting in neural processing [6], and hence, this simple system provides a model for studying its incidence in network dynamics. References [1] M. Abeles, Corticonics: Neural Circuits of the Cerebral Cortex, Cambridge University Press, Cambridge, 1991. [2] N. Brunel, Dynamics of networks of randomly connected excitatory and inhibitory spiking neurons, J. Physiol. (Paris) 94 (2000) 445–463. [3] M. Diesmann, M.-O. Gewaltig, A. Aertsen, Stable propagation of synchronous spiking in cortical neural networks, Nature 402 (1999) 529–533. [4] J.J. Eggermont, Is there a neural code? Neurosci. Biobehav. Rev. 22 (1998) 355–370. [5] G. Fortin, J. Champagnat, Spontaneous synaptic activities in rat nucleus tractus solitarius neurons in vitro, Brain Res. 630 (1993) 125–135.
262
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[6] L. Menendez de la Prida, N. Stollenwerk, J.V. Sanchez-Andres, Bursting as a source for predictability in biological neural network activity, Phys. D 110 (1997) 323–331. [7] J. Pham, K. Pakdaman, J. Champagnat, J.-F. Vibert, Activity in sparsely connected excitatory neural networks: e8ect of connectivity, Neural Networks 11 (1998) 415–434. [8] T.W. Troyer, K.D. Miller, Physiological gain leads to high ISI variability in a simple model of a cortical regular spiking cell, Neural Comput. 9 (1997) 971–983. [9] H.C. Tuckwell, Introduction to Theoretical Neurobiology, Cambridge University Press, Cambridge, 1988. [10] C. van Vreeswijk, L.F. Abbott, G.B. Ermentrout, When inhibition not excitation synchronizes neural 1ring, J. Comput. Neurosci. 1 (1994) 313–321. [11] C. van Vreeswijk, D. Hansel, Patterns of synchrony in neural networks with spike adaptation, Neural Comput. 13 (2001) 959–992. [12] J.-F. Vibert, F. Alvarez, E. Kosmidis, XNBC V9: A user friendly simulation and analysis tool for neurobiologists, Neurocomputing 38– 40 (2001) 1715–1723. [13] J.-F. Vibert, F. Alvarez, J. Pham, E8ect of transmission delays and noise in recurrent excitatory neural networks, BioSystems 48 (1998) 255–262. [14] J.-F. Vibert, K. Pakdaman, N. Azmy, Interneural delay modi1cation synchronizes biologically plausible neural networks, Neural Networks 7 (1994) 589–607. [15] J.A. White, Synchronization and oscillatory dynamics in heterogeneous, mutually inhibited neurons, J. Comput. Neurosci. 5 (1998) 5–16.