autovalori della matrice monodroma soddisfano |ρm| < 1, con m 6= t, l’orbita periodica `e
stabile. Se invece almeno uno dei moltiplicatori ha modulo maggiore di uno allora l’orbita periodica `e instabile. Tali risultati, con gli opportuni cambiamenti, si estendono anche al caso in cui i moltiplicatori non sono distinti [13].
A.4
Equazioni di fase
Supponiamo che il sistema dinamico ˙X = F(X) abbia un ciclo limite stabile Γ di periodo T e che sia soggetto ad una piccola perturbazione:
˙
X = F(X) + G(X, t). (A.11)
Dato che la caratterizzazione dinamica delle soluzioni di questo sistema di equazioni `e non banale, l’obbiettivo `e quello di semplificarlo a un sistema unidimensionale nella sola variabile di fase θ(X). Per ottenere tale riduzione si impone che per = 0 si abbia:
dθ(X)
dt = 1. (A.12)
dove X ∈ U e U `e il bacino di attrazione del ciclo limite Γ. Per la regola di derivazione delle funzioni composte dovr`a essere:
dθ(X) dt = ∂θ(X) ∂X · F(X) + ∂θ(X) ∂X · G(X, t) = 1 + ∂θ(X) ∂X · G(X, t), (A.13)
che `e una equazione differenziale alle derivate parziali per θ(X) e soddisfa la A.12 per = 0. In realt`a, in questa espressione compare ancora la dipendenza da X, mentre quello che si vuole `e una equazione per la sola θ. Si pu`o dimostrare [69] che, se la variabile θ, invece che nel generico X ∈ U , viene valutata all’intersezione tra il ciclo limite Γ e l’isocrona passante per X, la A.13 diventa:
dθ(X)
dt = 1 +
∂θ(XΓ(θ))
∂X · G(X
Γ(θ), t) + O(2), (A.14)
Bibliografia
[1] Kandel E.R., Schwartz J.H, Jessell T.M.: Fondamenti delle Neuroscienze e del comportamento, Casa Editrice Ambrosiana (1999).
[2] Thompson R.F.: Il Cervello, Introduzione alle Neuroscienze, Zanichelli (1997). [3] Cole K.S., Curtis H.J.: Electrical impedance of the squid giant axon during
activity, J. Gen. Physiol., vol. 22, pp. 649-670 (1939).
[4] Hodgkin A.L., Katz B.: The effect of sodium ions on the elecrical activity of the giant axon of the squid, J. Physiol., vol. 108, pp. 37-77 (1949).
[5] Hodgkin A.L., Huxley A.F.: A quantitative description of membrane current and its application to conduction and excitation in nerve, J. Physiol., vol. 117, pp. 500-544 (1952).
[6] FitzHugh R.: Impulses and physiological states in theoretical models of nerve membrane, Biophys. J., vol. 1, pp. 445-466 (1961).
[7] Nagumo J.S. et al.: An active pulse transmission line simulating nerve axon, Proc. IRE, vol. 50, pp. 2061-2070 (1962).
[8] Koch C.: Biophysics of Computation, Information processing in single neurons Oxford (1999).
[9] Morris C., Lecar H.: Voltage oscillations in the barnacle giant muscle fiber, Biophys. J., vol. 193, pp. 193-213 (1981).
[10] Ermentrout B.: Neural networks as spatio-temporal pattern-forming systems, Rep. Prog. Phys., vol. 61, pp. 353-430 (1998).
[11] Kopell N., Ermentrout B.: Symmetry and phase-locking in chains of weakly coupled oscillators, Comm.-Pure-Appl.-Math., vol. 39, n. 5, pp. 623-660 (1986). [12] Ermentrout B.: Type I membranes, phase resetting curves, and synchrony,
Neural Computation vol. 8(5), pp. 979-1001 (1996).
[13] Nayfeh H.A.,Balachandran B.: Applied Nonlinear Dynamics, Wiley-
Interscience (1995).
[14] Guckenheimer J., Holmes P.: Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields, Springer-Verlag.
[15] Glendinning P.: Stability, Instability and Chaos: an introduction to the theory of nonlinear differential equations, Cambridge.
[16] Strogatz S.H.: Nonlinear Dynamics and Chaos, Addison-Wesley (1995). [17] Crawford J.D.: Introduction to Bifurcation Theory, J. Reviews of Modern
Physics, vol. 63 (1991).
[18] Izhikevich E.M., Hoppensteadt F.C.: Weakly Connected Neural Networks, Springer (1997)
[19] Gray G.M., Singer W.: Stimulus-specific neuronal oscillations in orientation columns of rat visual cortex, PNAS, vol. 86, pp. 1698-1702 (1989).
[20] Singer W.: Search for coherence: a basic principle of cortical self-organization, Concept. in Neurosci., vol. 1, pp. 1-26 (1990).
[21] Buzsaki G., Draguhn A.: Neuronal oscillations in Cortical Networks, Science, vol. 304, pp. 1926-1929, (2004).
[22] Salinas E., Sejnowski T.J.: Correlated Neuronal Activity and the flow of neural information, Nature Reviews Neuroscience, vol. 2, pp. 539-550, (2001).
[23] Engel A.K., Singer W.: Temporal binding and the neural correlates of awareness, Trends in Cognitive Sciences, vol. 5, pp. 16-25, (2001).
[24] Reppert S.M.: A Clockwork explosion, Neuron, vol. 21, pp. 1-4, (1998). [25] Gross J.: The neural basis of intermittent motor control in humans, PNAS,
vol. 99, pp. 2299-2302, (2002).
[26] McCormick D.A.,Contreras D.: On the cellular and networks bases of epileptic seizures, Ann. Rev. Physiol., vol. 63, pp. 815-846, (2001).
[27] de Curtis M., Avanzini G.: Interictal Spikes in Focal Epileptogenesis, Progress in Neurobiology, vol. 63, pp. 541-567, (2001).
[28] Williams L.M. et al.: Synchronous gamma activity: a review and contribution to an integrative neuroscience model of schizofrenia, Brain Research Reviews, vol. 41, pp. 57-78, (2003).
[29] McCarley R.W.: Neural synchrony indexes disordered perception and
cognition in schizofrenia, PNAS, vol. 101, pp. 17288-17293, (2004).
[30] Schnitzler A.,Gross J.: Normal and pathological oscillatory communication in the brain, Nature Reviews Neuroscience, vol. 6, pp. 285-296, (2005).
BIBLIOGRAFIA 119
[31] Winfree A.T.: The Geometry of Biological Time, NewYork, Springer (2001).
[32] Izhikevich E.M.: Dynamical Systems in Neuroscience: The geometry of
Excitability and Bursting, MIT press (2005).
[33] Sanders J.A.,Verhulst F.: Averaging Methods in Nonlinear Dynamical
Systems, Springer-Verlag, N.Y. (1985).
[34] Whittington M.A., Traub R.D., Jefferys J.G.R.: Synchronised oscillations in interneuron networks driven by metabotropic glutamate receptor activation, Nature, vol. 373, pp. 612-615 (1995)
[35] Traub R.D. et al.: Analysis of gamma-rhythms in the rat hippocampus in vitro and in vivo, J. Physiology, vol. 493, pp. 471-484 (1996)
[36] Traub R.D., Jefferys J.G.R., Whittington M.A.: Simulations of gamma- rhythms in networks of interneurons and pyramidal cells, J. Computational Neuroscience, vol. 4, pp. 141-150 (1997)
[37] Whittington M.A. et al.: Inhibition-based rhythms: experimental and math- ematical observations on network dynamics, Int. J. Psychophysiology, vol. 38, pp. 315-336 (2000)
[38] Van Vreeswijk C., Abbott L.F., Ermentrout B.: When inhibition not exci- tation synchronizes neural firing, J. Computational Neuroscience, vol. 1, pp. 313-321 (1994)
[39] Wang X.J., Buzsaki G.: Gamma oscillation by synaptic inhibition in a hip- pocampal interneuronal network model, J. Neuroscience, vol. 16(20), pp. 6402-6413 (1996)
[40] White J.A. et al.: Synchronization and oscillatory dynamics in heterogeneous, mutually inhibited neurons, J. Computational Neuroscience, vol. 5, pp. 5-16 (1998)
[41] LeBeau F.E.N. et al.: The role of electrical signalling via gap-junctions in the generation of fast network oscillations, Brain Research Bulletin, vol 62, pp. 3-13, (2003)
[42] Kawaguchi Y., Kubota Y.: GABAergic cell subtypes and theyr synaptic connections in rat frontal cortex, Cerebral Cortex, vol. 7, pp. 476-486 (1997) [43] Markram H. et al.: Interneurons of the neocortical inhibitory system, Nature
Reviews Neuroscience, vol. 5, pp. 793-807 (2004)
[44] Galarreta M., Hestrin S.: A network of fast-spiking cells in the cortex connected by electrical synapses, Nature, vol. 402, pp. 72-75 (1999)
[45] Gibson J.R., Beierlein M., Connors B.: Two networks of electrically coupled inhibitory neurons in neocortex, Nature, vol. 402, pp. 75-79 (1999)
[46] Galarreta M., Hestrin S.: Electrical synapses between GABA-releasing interneurons, Nature Rev. Neuroscience, vol. 2, pp. 425-433 (2001)
[47] Deans M.R. et al.: Synchronous activity of inhibitory networks in neocortex requires electrical synapses containing connexin36, Neuron, vol. 31, pp. 477- 485 (2001)
[48] Ermentrout B., Kopell N.: Frequency plateaus in a chain of weakly coupled oscillators with random frequencies, Math. Biol., vol. 22, pp. 1-9 (1984) [49] Abbott L.F.: A network of oscillators, J. Phys. A, vol. 23, p. 3835 (1990) [50] Lewis T.J., Rinzel J.: Dynamics of spiking neurons connected by both in-
hibitory and electrical coupling, J. Computational Neuroscience, vol. 14, pp. 283-309 (2003)
[51] Pfeuty B., Mato G., Golomb D., Hansel D.: Electrical synapses and synchrony: the role of intrinsic currents, J. Neuroscience, vol. 23(15), pp. 6280-6294 (2003) [52] Kopell N., Ermentrout B.: Chemical and electrical synapses perform comple- mentary roles in the synchronization of interneuronal networks, PNAS, vol. 26, pp. 15482-15487 (2004)
[53] Bem T., Rinzel J.: Short Duty Cycle Destabilizes a Half-Center Oscillator, But Gap Junctions Can Restabilize the Anti-Phase Pattern, J. Neurophysiol., vol. 91, pp. 693-703 (2004)
[54] Pfeuty B., Mato G., Golomb D., Hansel D.: The Combined Effects of In- hibitory and Electrical Synapses in Synchrony, Neural Computation, vol. 17, pp. 633-670 (2005)
[55] Skinner F.K., Bazzazi H., Campbell S.A.: Two-Cell to N-Cell Heterogeneous, Inhibitory Networks: Precise Linking of Multistable and Coherent Properties, J. Computational Neuroscience, vol. 18, pp. 343-352 (2005)
[56] Erisir A. et al.: Function of specific K+ channels in sustained high-frequency
firing of fast-spiking neocortical interneurons, J. Neurophys., vol. 82, pp. 2476- 2489, (1999)
[57] Bartos M., Vida I. et al.: Rapid signaling at inhibitory synapses in a dentate gyrus interneuron network, J. Neuroscience, vol. 21(8), pp. 2687-2698 (2001) [58] Bartos M., Vida I. et al.: Fast synaptic inhibition promotes synchronized
gamma oscillations in hippocampal interneuron networks, PNAS, vol. 99(20), pp. 13222-13227 (2002)
BIBLIOGRAFIA 121
[59] Martina M., Jonas P.: Functional differences in N a+ channel gating be-
tween fast-spiking interneurones and principal neurones of rat hippocampus, J. Physiology, vol. 505.3, pp. 593-603 (1997)
[60] Lewis T.J., Rinzel J.: Dendritic effects in networks of electrically coupled fast-spiking interneurons, Neurocomputing, vol. 58-60, pp. 145-150, (2004) [61] Bacci A. et al.: Major differences in inhibitory synaptic transmission onto two
neocortical interneuron subclasses, J. Neuroscience, vol. 23, pp. 9664-9674 (2003)
[62] Martina M. et al.: Functional and molecular differences between voltage- gated K+ channels of fast-spiking interneurons and pyramidal neurons of rat hippocampus, J. Neuroscience, vol. 18(20), pp. 8111-8125 (1998)
[63] Lien C.C., Jonas P.: Kv3 potassium conductance is necessary and kinetical- ly optimized for high-frequency action potential generation in hippocampal interneurons, J. Neuroscience, vol. 23(6), pp. 2058-2068 (2003)
[64] Williams T.L., Bowtell G.: The calculation of frequency-shift functions for chains of coupled oscillators, with application to a network model of the lam- prey locomotor pattern generator, J. Computational Neuroscience, vol. 4, pp. 47-55 (1997)
[65] Di Garbo A.,Panarese A., Chillemi S.: Gap junctions promote synchronous activities in a network of inhibitory interneurons, BioSystems, vol. 79, pp. 91-99 (2005)
[66] Galarreta M., Hestrin S.: Spike transmission and synchrony detection in networks of GABAergic interneurons, Science, vol. 292, pp. 2295-2299 (2001) [67] Destexhe A. et al.: The discharge variability of neocortical neurons during
high-conductance states, Neuroscience, vol. 119, pp. 855-873 (2003)
[68] Greiner A. et al.: Numerical Integration of stochastic differential equations, J. Statistical Physics, vol. 51, pp. 95-108 (1988)
[69] Brown E., Moehlis J., Holmes P.: On the phase reduction and response dynam- ics of neural oscillator populations, Neural Computation, vol. 16, pp. 673-715 (2004)
Ringraziamenti
Al termine di questo lavoro, il mio grazie pi`u grande va a Dio, che pi`u di ogni altro ha saputo darmi forza, infondermi coraggio, insegnarmi a sperare.
Un sincero ringraziamento va anche ad Angelo Di Garbo, che ha reso possibile questo lavoro e che vi ha partecipato con dedizione e pazienza, a Santi Chillemi e a Michele Barbi, per i loro preziosi consigli.
Un grazie speciale va poi alla mia fidanzata, Prisca, che non mi ha mai lasciato solo e che con il suo sorriso `e capace di farmi dimenticare ogni fatica. Grazie, infine, alla mia famiglia, che ha sempre creduto in me e mi ha sostenuto incessantemente e alla famiglia di Prisca, che mi ha accolto come uno di loro.