document: about ten years ago , a peculiar dynamical phenomenon was discovered in populations of identical phase oscillators : under nonlocal symmetric coupling , the coexistence of coherent ( synchronized ) and incoherent oscillators was observed @xcite . 
   this highly counterintuitive phenomenon was given the name chimera state after the greek mythological creature made up of different animals @xcite . since 
   then the study of chimera states has been the focus of extensive research in a wide number of models , from kuramoto phase oscillators @xcite to periodic and chaotic maps @xcite , as well as stuart - landau oscillators @xcite . 
   the first experimental evidence of chimera states was found in populations of coupled chemical oscillators as well as in optical coupled - map lattices realized by liquid - crystal light modulators @xcite . 
   recently , moreover , martens and coauthors showed that chimeras emerge naturally from a competition between two antagonistic synchronization patterns in a mechanical experiment involving two subpopulations of identical metronomes coupled in a hierarchical network @xcite .    in the context of neuroscience 
   , a similar effort has been undertaken by several groups , since it is believed that chimera states might explain the phenomenon of unihemispheric sleep observed in many birds and dolphins which sleep with one eye open , meaning that one hemisphere of the brain is synchronouns whereas the other is asynchronous @xcite . 
   the purpose of this paper is to make a contribution in this direction , by identifying for the first time a variety of single and multi - chimera states in networks of non - locally coupled neurons represented by hindmarsh  rose oscillators . 
   recently , multi - chimera states were discovered on a ring of nonlocally coupled fitzhugh - nagumo ( fhn ) oscillators @xcite . 
   the fhn model is a 2dimensional ( 2d ) simplification of the physiologically realistic hodgkin - huxley model @xcite and is therefore computationally a lot simpler to handle . 
   however , it fails to reproduce several important dynamical behaviors shown by real neurons , like rapid firing or regular and chaotic bursting . 
   this can be overcome by replacing the fhn with another well - known more realistic model for single neuron dynamics , the hindmarsh - rose ( hr ) model @xcite , which we will be used throughout this work both in its 2d and 3d versions .    in section 2 of this paper 
   , we first treat the case of 2d - hr oscillators represented by two first order ordinary differential equations ( odes ) describing the interaction of a membrane potential and a single variable related to ionic currents across the membrane under periodic boundary conditions . 
   we review the dynamics in the 2d plane in terms of its fixed points and limit cycles , coupling each of the @xmath0 oscillators to @xmath1 nearest neighbors symmetrically on both sides , in the manner adopted in @xcite , through which chimeras were discovered in fhn oscillators . 
   we identify parameter values for which chimeras appear in this setting and note the variety of oscillating patterns that are possible due to the bistability features of the 2d model . 
   in particular , we identify a new `` mixed oscillatory state '' ( mos ) , in which the desynchronized neurons are uniformly distributed among those attracted by a stable stationary state . 
   furthermore , we also discover chimera  like patterns in the more natural setting where only the membrane potential variables are coupled with @xmath2 of the same type . 
   next , we turn in section 3 to the more realistic 3d - hr model where a third variable is added representing a slowly varying current , through which the system can also exhibit bursting modes . here , we choose a different type of coupling where the two first variables are coupled symmetrically to @xmath2 of their own kind and observe only states of complete synchronization as well as mos in which desynchronized oscillators are interspersed among neurons that oscillate in synchrony . however , when coupling is allowed only among the first ( membrane potential ) variables chimera states are discovered in cases where spiking occurs within sufficiently long bursting intervals . 
   finally , the paper ends with our conclusions in section 4 . 
   following the particular type of setting proposed in @xcite we consider @xmath0 nonlocally coupled hindmarsh - rose oscillators , where the interconnections between neurons exist with @xmath3 nearest neighbors only on either side as follows : @xmath4 \label{eq:01 } \\ 
   \dot y_k & = & 1 - 5x_k^2-y_k+ \frac{\sigma_y}{2r}\sum_{j = k - r}^{j = k+r } [ b_{yx}(x_j - x_k)+b_{yy}(y_j - y_k ) ] . 
   \label{eq:02 }     \end{aligned}\ ] ] in the above equations @xmath5 is the membrane potential of the @xmath6-th neuron , @xmath7 represents various physical quantities related to electrical conductances of the relevant ion currents across the membrane , @xmath8 , @xmath9 and @xmath10 are constant parameters , and @xmath11 is the external stimulus current . each oscillator is coupled with its @xmath12 nearest neighbors on both sides with coupling strengths @xmath13 . 
   this induces nonlocality in the form of a ring topology established by considering periodic boundary conditions for our systems of odes . as in @xcite , 
   our system contains not only direct @xmath14 and @xmath15 coupling , but also cross - coupling between variables @xmath16 and @xmath17 . 
   this feature is modeled by a rotational coupling matrix : @xmath18 depending on a coupling phase @xmath19 . in 
   what follows , we study the collective behavior of the above system and investigate , in particular , the existence of chimera states in relation to the values of all network parameters : @xmath0 , @xmath3 , @xmath19 , @xmath20 and @xmath21 . 
   more specifically , we consider two cases : ( @xmath22 ) direct and cross - coupling of equal strength in both variables ( @xmath23 ) and ( @xmath24 ) direct coupling in the @xmath16 variable only ( @xmath25 , @xmath26 ) . similarly to @xcite we shall use initial conditions randomly distributed on the unit circle ( @xmath27 ) .     at @xmath28 of eqs . 
   ( [ eq:01]),([eq:02 ] ) ( left ) and selected time series ( right ) for @xmath29 . 
   ( a ) @xmath30 , ( b ) @xmath31 and ( c ) @xmath32 . @xmath33 and @xmath34.,scaledwidth=70.0% ]    typical spatial patterns for case ( @xmath22 ) are shown on the left panel of fig . 
   [ fig : fig1 ] , where the @xmath16 variable is plotted over the index number @xmath6 at a time snapshot chosen after a sufficiently long simulation of the system . in this figure 
   the effect of different values of the phase @xmath19 is demonstrated while the number of oscillators @xmath0 and their nearest neighbors @xmath3 , as well as the coupling strengths @xmath23 are kept constant . for example , for @xmath30 ( fig . 
   [ fig : fig1](a ) ) an interesting novel type of dynamics is observed that we shall call `` mixed oscillatory state '' ( mos ) , whereby nearly half of the @xmath5 are attracted to a fixed point ( at this snapshot they are all at a value near @xmath35 ) , while the other half are oscillating interspersed among the stationary ones . from the respective time series ( fig .  [ 
   fig : fig1](a ) , right ) it is clear that the former correspond to spiking neurons whereas the latter to quiescent ones . 
   this interesting mos phenomenon is due to the fact that , in the standard parameter values we have chosen , the uncoupled hr oscillators are characterized by the property of _ 
   bistability_. clearly , as shown in the phase portrait of fig . 
   [ fig : fig2 ] , each oscillator possesses three fixed points : the leftmost fixed point is a stable node corresponding to the resting state of the neuron while the other two correspond to a saddle point and an unstable node and are therefore repelling . for @xmath11 ( which is the case here ) 
   a stable limit cycle also exists which attracts many of the neurons into oscillatory motion , rendering the system bistable and producing the dynamics observed in fig . 
   [ fig : fig1](a ) . 
   now , when a positive current @xmath36 is applied , the @xmath16-nullcline is lowered such that the saddle point and the stable node collide and finally vanish . in this case 
   the full system enters a stable limit cycle associated with typical spiking behaviour . 
   similar complex patterns including mos and chimeras have been observed in this regime as well .    . 
   three fixed points coexist with a stable limit cycle.,scaledwidth=40.0% ]    for @xmath31 , on the other hand , there is a `` shift '' in the dynamics of the individual neurons into the spiking regime , as seen in fig .  [ 
   fig : fig1](b ) ( right ) . 
   the corresponding spatial pattern has a wave - like form of period 2 . 
   then , for @xmath32 a classical chimera state with two incoherent domains is observed ( ( fig . 
   [ fig : fig1](c ) , left ) . diagonal coupling ( @xmath37 , @xmath38 ) is , therefore , identified as the necessary condition to achieve chimera states . by contrast , it is interesting to note that in nonlocally coupled fitz - hugh nagumo oscillators @xcite it has been shown both analytically and numerically that chimera states occur for _ off - diagonal _ coupling .    by decreasing the range of coupling @xmath3 and increasing the system size @xmath0 , chimera states occur with multiple domains of incoherence and coherence for @xmath32 ( fig . 
   [ fig : fig3](c , d ) ) , and , accordingly , periodic spatial patterns with larger wave numbers arise for @xmath31 , as seen in fig . 
   [ fig : fig3](a , b ) . 
   this is in agreement with previous works reported in @xcite . 
   next we consider the case ( @xmath24 ) where the coupling term is restricted to the @xmath16 variable . 
   this case is important since incorporating the coupling in the voltage membrane ( @xmath16 ) alone is more realistic from a biophysiological point of view . in fig . 
   [ fig : fig4 ] spatial plots ( left ) and the corresponding @xmath39-plane ( right ) for increasing coupling strength are shown . 
   chimera states ( fig . 
   [ fig : fig4 ] ( b , c ) ) are observed as an intermediate pattern between desynchronization ( fig . 
   [ fig : fig4 ] ( a ) ) and complete synchronization ( fig . 
   [ fig : fig4 ] ( d ) ) .     of eqs . 
   ( [ eq:01]),([eq:02 ] ) at @xmath28 for @xmath23 . 
   @xmath40 ( top ) and @xmath41 ( bottom ) whereas @xmath31 ( left ) and @xmath32 ( right ) . in ( a ) and ( b ) @xmath42 , in ( c ) @xmath43 , and in ( d ) @xmath44.,scaledwidth=50.0% ]     ( left ) and in the @xmath45-plane ( right ) of eqs .  ( [ eq:01]),([eq:02 ] ) at @xmath28 for @xmath25 , @xmath33 and @xmath34 . ( 
   a ) @xmath46 , ( b ) @xmath47 , ( c ) @xmath48 and ( d ) @xmath49.,scaledwidth=40.0% ] 
   in order to complete our study of the hindmarsh - rose model we shall consider , in this section , its three - dimensional version . the corresponding equations read :    @xmath50    the extra variable @xmath51 represents a slowly varying current , which changes the applied current @xmath52 to @xmath53 and guarantees firing frequency adaptation ( governed by the parameter @xmath54 ) as well as the ability to produce typical bursting modes , which the 2d model can not reproduce . 
   parameter @xmath55 controls the transition between spiking and bursting , parameter @xmath56 determines the spiking frequency ( in the spiking regime ) and the number of spikes per bursting ( in the bursting regime ) , while @xmath57 sets the resting potential of the system . 
   the parameters of the fast @xmath58 system are set to the same values used in the two - dimensional version ( @xmath8 , @xmath10 ) and typical values are used for the parameters of the @xmath51-equation ( @xmath59 , @xmath60 , @xmath61 ) . 
   the 3d hindmarsh - rose model exhibits a rich variety of bifurcation scenarios in the @xmath62 parameter plane @xcite . 
   thus , we prepare all nodes in the spiking regime ( with corresponding parameter values @xmath9 and @xmath63 ) and , as in section 2 , we use initial conditions randomly distributed on the unit sphere ( @xmath64 ) .    first let us consider direct coupling in both variables @xmath16 and @xmath17 and vary the value of the equal coupling strengths @xmath23 , while @xmath0 and @xmath3 are kept constant . 
   naturally , the interaction of @xmath0 spiking neurons will lead to a change in their dynamics , as we discuss in what follows . for low values of @xmath65 
   we observe the occurrence of a type of mos where nearly half of the neurons spike regularly in a synchronous fashion , while the rest are unsynchronized and spike in an irregular fashion . 
   this is illustrated in the respective time series in the right panel of fig . 
   [ fig : fig5](a ) at higher values of the coupling strength the the system is fully synchronized ( fig . 
   [ fig : fig5](b ) ) .     of eqs . 
   ( [ eq:03])-([eq:05 ] ) at @xmath66 ( left ) and selected time series ( right ) for : ( a ) @xmath67 and ( b ) @xmath68 . @xmath33 and @xmath34.,scaledwidth=50.0% ] 
   next we check the case of coupling in the @xmath16 variable alone ( @xmath69 , @xmath25 ) . 
   figure  [ fig : fig6 ] displays characteristic synchronization patterns obtained when we increase @xmath20 ( left panel ) and selected time series ( right panel ) . at low values of the coupling strength 
   all neurons remain in the regular spiking regime and desynchronization alternates with complete synchronization as @xmath20 increases ( fig . 
   [ fig : fig6](a , b , c ) ) . for intermediate values of the coupling strength , 
   chimera states with one incoherent domain are to be observed . 
   these are associated with a change in the dynamics of the individual neurons , which now produce irregular bursts ( fig . 
   [ fig : fig6](d ) ) . 
   the number of spikes in each burst increases at higher @xmath20 values and the system is again fully synchronized ( fig . 
   [ fig : fig6](e ) ) . 
   extensive simulations show that the chimera states disappear and reappear again by varying @xmath20 which is most likely due to the system s multistability and sensitive dependence on initial conditions .     of eqs . 
   ( [ eq:03])-([eq:05 ] ) at @xmath66 ( left ) and selected time series for @xmath25 , @xmath33 and @xmath34 . ( a ) 
   @xmath70 , ( b ) @xmath71 , @xmath72 , ( d ) @xmath73 , and ( e ) @xmath74 . , 
   in this paper , we have identified the occurrence of chimera states for various coupling schemes in networks of 2d and 3d hindmarsh - rose models . 
   this , together with recent reports on multiple chimera states in nonlocally coupled fitzhugh - nagumo oscillators , provide strong evidence that this counterintuitive phenomenon is very relevant as far as neuroscience applications are concerned . 
   chimera states are strongly related to the phenomenon of synchronization . during the last years 
   , synchronization phenomena have been intensely studied in the framework of complex systems @xcite . moreover , it is a well - established fact the key ingredient for the occurrence of chimera states is nonlocal coupling . 
   the human brain is an excellent example of a complex system where nonlocal connectivity is compatible with reality . 
   therefore , the study of chimera states in systems modelling neuron dynamics is both significant and relevant as far as applications are concerned . 
   moreover , the present work is also important from a theoretical point of view , since it verifies the existence of chimera states in coupled bistable elements , while , up to now , it was known to arise in oscillator models possessing a single attracting state of the limit cycle type . 
   finally , we have identified a novel type of mixed oscillatory state ( mos ) , in which desynchronized neurons are interspersed among those that are either stationary or oscillate in synchrony . as a continuation of this work , it is very interesting to see whether chimeras and mos states appear also in networks of _ populations _ of hindmarsh - rose oscillators , which are currently under investigation . 
   the authors acknowledge support by the european union ( european social fund  esf ) and greek national funds through the operational program `` education and lifelong learning '' of the national strategic reference framework ( nsrf ) - research funding program : thales . investing in knowledge society through the european social fund . 
   funding was also provided by ninds r01 - 40596 . 
   hagerstrom , a.  m. , thomas , e. , roy , r. , hvel , p. , omelchenko , i. & schll , e. [ 2012 ] `` chimeras in coupled - map lattices : experiments on a liquid crystal spatial light modulator system '' , _ nature physics _ * 8 * , 658 . 
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summary: we have identified the occurrence of chimera states for various coupling schemes in networks of two - dimensional and three - dimensional hindmarsh - rose oscillators , which represent realistic models of neuronal ensembles . 
   this result , together with recent studies on multiple chimera states in nonlocally coupled fitzhugh - nagumo oscillators , provide strong evidence that the phenomenon of chimeras may indeed be relevant in neuroscience applications . moreover , our work verifies the existence of chimera states in coupled bistable elements , whereas to date chimeras were known to arise in models possessing a single stable limit cycle . 
   finally , we have identified an interesting class of mixed oscillatory states , in which desynchronized neurons are uniformly interspersed among the remaining ones that are either stationary or oscillate in synchronized motion .