Rhythmic activity in neuronal ensembles in the presence of conduction delays

Transcription

1 Rhythmc actvty n neuronal ensembles n the presence of conducton delays Crstna Masoller Carme Torrent, Jord García Ojalvo Departament de Fsca Engnyera Nuclear Unverstat Poltecnca de Catalunya, Terrassa, Span GABA 2st Scentfc Workshop Pars, France, March 29

2 Motvaton How do dstant neurons synchronze they actvty when there s a large number of cells wth complex nteractons wth heterogeneous conducton delays n a stochastc envronment?

3 Outlne ntroducton: bologcal detals requred for modelng nterconnected neurons Neurons Synaptc connectons Conducton delays Neuronal networks Sngle neuron wth recurrent connecton & globally coupled neurons - nfluence of couplng strength - of delay tme - of nose Summary, Conclusons & Future Work

4 Neurons Core components of the bran, the vertebrate spnal cord, the nvertebrate ventral nerve cord, and the perpheral nerves Dendrte Soma Axon termnal Types: - sensory neurons (respond to touch, sound, lght, etc). - motor neurons (receve sgnals from the bran and cause muscle contractons, etc.) -nter-neurons (connect neurons to other neurons) Nucleo Axon

5 Sngle neuron model The neuron s broken nto many small compartments, each of whch conssts of a smple RC crcut. The resstances (called conductances) are a consequence of onc channels n the neuron membrane. The current through one of these channels s modeled as ohmc: After Braun et al, JBC 1998 appled

6 Neurons communcate by synapses Chemcal synapses: spke-medated communcaton The spke travels down the axon whch termnates n a seres of regons called synaptc termnals. n the synaptc termnals the potental causes the release of a neurotransmtter. Ths transmtter causes channels to open up and current to pass through the dendrtes. The current can be depolarzng (exctatory) o hyperpolarzng (nhbtory).

7 Electrcal synapses An electrcal synapse s a mechancal and electrcally conductve lnk between two neurons that s formed at a narrow gap between the pre- and postsynaptc cells, also known as a gap juncton. At gap junctons, the neurons approach wthn 3.5 nm of each other (<< than the 2-4 nm dstance that separates neurons at chemcal synapses). Gap junctons are connectons between cells whch allow passage of small molecules and electrc current. Several structures n the bran have been shown to contan electrcally coupled neurons (the vestbular nucleus, the nucleus of trgemnal nerve, the nferor olvary nucleus, and the Ventral Tegmental Area).

8 Electrcal synapses vs. chemcal synapses Compared to chemcal synapses, electrcal synapses conduct mpulses faster, but unlke chemcal synapses they do not have gan (the sgnal n the post synaptc neuron s always smaller than that of the orgnatng neuron). A characterstc of electrcal synapses s that they are usually bdrectonal. Gap-junctons are expected to play a role n synchronzng neuronal actvty, n contrast to chemcal synapses, that are the mechansm for forward transmsson of nformaton.

9 Axonal conducton delays The longest axon of a human motor neuron can be over a meter long, reachng from the base of the spne to the toes. Sensory neurons have axons that run from the toes to the dorsal columns, over 1.5 meters n adults. Acton potentals travel along the axon at speeds of 1-1 m/s. n synaptcally-coupled neurons there can be non-neglegble delays due to the dstance travelled down the axon before the frng releases a neurotransmtter. n electrcally-coupled neurons delays are expected to be shorter.

10 Neuronal networks Employed to descrbe space-tme actvty n real neural systems Very dfferent types of models have been proposed Here we use a very smple model wth two strong smplfcatons: 1) the post-synaptc neuron sees a potental that has the same shape as n the presynaptc neuron that caused t. The sgn and the ampltude can be dfferent. 2) The conducton delay s the same for all neurons.

11 Globally coupled neurons C dv dt Na, K, sd, sr, l, N N j 1 V j ( t ) 1... N Synchronzed actvty: V ( t) Vj( t), j, t C dv dt Na, K, sd, sr, l, V ( t ) 1... N To understand the synchronzed actvty of an ensemble of delayed-coupled neurons, s mportant to understand the behavor of a sngle neuron wth a recurrent connecton (delayed feedback).

12 Hodgkn-Huxley model wth delayed feedback For nhbtory feedback and certan delay tmes, dfferent ntal condtons result n dfferent frng patterns Foss, Longtn, Mensour and Mlton, PRL 1996 Foss and Mlton, Neurophysology 2

13 nfluence of delayed feedback n a neuron model dsplayng sub-threshold oscllatons Braun et al model, JBC 1998 C M dv dt Na K sd sr l V ( t ) 2D ( t) Determnstc model (D=) : we chose parameters such that the neuron dsplays, n the absence of feedback ( =), sub-threshold oscllatons of perod T We fnd: feedback-nduced spkes for < and certan delay values ( nt ), no spkes for > 2 o C 25 o C 3 o C 35 o C

14 S (determnstc model, <) Delay tme (n unts of T ) Masoller, Torrent, García-Ojalvo, PRE 28

15 Frng rate (determnstc model, <) Wthout nose: the spke pattern s controlled by the feedback delay tme

16 Membrane potental (mv) Mult-stablty (determnstc model, <) At certan delay values dfferent patterns arse from dfferent random ntal condtons /T =8.5 S: 2T, 4T S: 2T -5 S: 2T, 3T Tme /T

17 nterplay of nose and sub-threshold actvty Wthout recurrent connecton ( =) no nose wth nose 2 o C 25 o C 3 o C 35 o C Braun et al, PRE 2

18 nterplay of nose and delayed feedback no nose < > wth nose Feedback Nose suppressed spkes spkes Masoller, Torrent, García-Ojalvo, PRE 28

19 Spkes when there s nose and feedback =: nose-nduced spkes =-.1, =T /2 =-.1, =T =.1, =T /2 =.1, =T

20 nterpretaton: nfluence of feedback on a lmt cycle oscllator =.1 = -.1

21 Membrane potental and Mean Feld Globally coupled neurons C dv dt Na, K, sd, sr, l, N N j 1 V j ( t ) 1... N V(t) 1 N N j 1 V (t) j < Ampltude of the oscllatons of the mean feld (black) & 5 neurons (symbols) /T

22 Synchronzaton and clusterng are controlled by the delay tme of the couplng < 2 /T =6.6 2 /T = /T =6.8 2 /T =7. V (mv) /T =7.3 2 /T = Tme[T ]

23 Ampltude mean feld Multstablty 8 7 Ampltude of the oscllatons of the mean feld for 8 dfferent random ntal condtons /T

24 nfluence of the system sze ( >) Ampltude of the mean feld oscllatons

25 For negatve couplng Ampltude of the mean feld oscllatons Ampltude of the neuronal oscllatons

26 nterpretaton: lmt-cycle oscllators wth global delayed couplng Ampltude of the oscllatons of the mean feld (red crcles) & of 5 oscllators (black dots)

27 nterpretaton (): phase-flp bfurcaton n two delayed coupled oscllators Prasad et al, PRE 26, Chaos 28

28 Summary and Conclusons We studed the dynamcs of a neuron wth delayed feedback representng a recurrent connecton. The neuron dsplays sub-threshold oscllatons; the feedback was ncluded as a tme-delayed lnear term. Weak negatve feedback enhance the oscllaton ampltude, nducng threshold-crossngs and frng actvty for certan wndows of delay values. n these wndows the frng pattern s controlled by the delay. Multstablty of solutons s found for certan delays. We also studed a small ensemble of neurons globally coupled through ther delayed mean feld. We found synchronzaton, out of phase oscllatons, and clusterng, wth the clusters exhbtng ant-phased oscllatons. As the delay vares the dfferent regmes repeat themselves n a perodc sequence (synchronzaton, out of phase behavor, clusters, and synchronzaton) wth a perodcty approxmately equal to T.