there have been numerous investigations of the effects of noise in neurobiological dynamical systems , for both single neurons and neural networks .
a large number of interesting noise - related phenomena have been analyzed , including synchronization , stochastic resonance , and coherence resonance ( reviewed in lindner et al .
most often , noise leads to increased responses ( stein et al . 2005 ) and sometimes to the phenomenon of stochastic resonance in which a maximum occurs in a response variable at a particular noise strength ( wiesenfeld and moss 1995 ; gammaitoni et al . 1998 ; badzey and mohanty 2005 ) .
this phenomenon has been demonstrated in neuronal systems , particularly , but not exclusively , in the case of periodic inputs ( gang et al . 1993 ; collins et al . 1996 ; longtin 1993 ; nozaki et al . 1999 ) .
2006 ; sim and forger 2007 ) and in neuronal models ( gutkin et al . 2008 ) that noise can subdue or turn off repetitive neuronal activity .
in this article , we report further investigations of such properties in hodgkin  huxley single neurons .
we find that there is a tuning effect of noise that has the opposite character to stochastic resonance , and thus , it might be called inverse stochastic resonance .
we argue that these phenomena will occur generically in neural and other dynamical systems that exhibit a specific kind of bistability between a stable limit cycle and a stable focus . in the central nervous system ,
the responses of many types of neurons to input currents , whether injected or synaptic , have been much investigated experimentally and theoretically ( mccormick et al .
they receive input signals from many other neurons through thousands of excitatory and inhibitory synapses at unpredictable or random times ( destexhe et al .
2001 ) . in order for a neuron to send out a signal , called an action potential ( bean 2007 ) , it must receive sufficient net excitation ( over inhibition ) in a small enough time interval so that the current or voltage distribution in the cell passes through some threshold condition .
once the threshold is reached , self - exciting processes lead to the emission of an action potential .
huxley model ( hodgkin and huxley 1952 ) , which is capable of reproducing spiking behavior similar to those of many real neurons and has often been employed in investigations of the effects of noise on neuronal dynamics ( brown et al .
1999 ; schmid et al . , 2004 ; tuckwell 2005 ; torcini et al .
we employ two commonly used different models for the input to the cell , which are described as  current - driven  and  conductance - driven 
( destexhe et al . 2001 ; tiesinga et al . 2000 ) . in the first of these ,
the input current is additive , being of the form i(t )  =    +  w(t ) , where t is time ,  and  are constant , and w(t ) is standard gaussian white noise
. such an input may represent somatic current injection by microelectrode in the laboratory . in the second model , which is closer to the nature of synaptic stimulation as it occurs in the nervous system , the driving current is of the form i(t ) 
=  g(t)(ve    v(t ) ) , where v(t ) is the membrane depolarization , ve  >  0 is the reversal potential for synaptic excitation ( tuckwell 1988 ) , and g(t ) satisfies a first - order differential equation containing a noise amplitude e and a conductance parameter ge , along with a time constant ( destexhe et al .
if there is inhibition , there is a similar term involving the inhibitory reversal potential , but here , we consider excitation only .
all the model equations for both kinds of input current and the relevant parameter values are given in the appendix .
we first demonstrate the unusual inhibitory effect of noise in the case of a hodgkin  huxley neuron with a current - based input when the mean current is just greater than the critical value for repetitive firing .
figure  1a shows the voltage responses of the model neuron in such a case with various noise levels .
initially , the neuron is taken to be at rest , which , by definition ( see hodgkin and huxley 1952 ) , implies that the depolarization is zero . the applied signal has a mean of strength  and a noisy component of amplitude . without noise ( top left record ) , there is , for this value of the steady input ,   =  6.6 , a repetitive stream of output spikes , there being eight in the time period duration of 150  ms shown . adding noise makes the output sequence of spikes irregular .
extremely weak noise naturally has little effect , but slightly larger amounts can have a significant effect on the neuron s spiking activity . moreover
, moderate amounts of noise can actually arrest the spiking for a long time . in the examples shown , a noise level of   =  0.2 can halt the firing of action potentials after three spikes , and a somewhat larger noise level of   =  0.5 here stops the spiking after just one spike . when the noise level is turned up to   =  2 ,
more spikes are emitted , there being 9 in the trial shown . in fig . 
n ( hodgkin and huxley 1952 ) , are shown . these phase - space diagrams are useful in understanding the effects of noise of various levels , as discussed below . 

the mean input current density is 6.6  a / cm and the effects of noise of various magnitudes ,  , are shown .
b orbits of voltage vs potassium activation variable corresponding to the plots of a. the limit cycle is clearly seen in the noise - free case and the manner in which small noise ,   =  0.2 , and intermediate noise ,   =  0.5 may switch the orbit away from the limit cycle . when   =  2 , the orbits are close to the noise - free case a showing voltage trajectories with spikes for a  current - driven 
the mean input current density is 6.6  a / cm and the effects of noise of various magnitudes ,  , are shown .
b orbits of voltage vs potassium activation variable corresponding to the plots of a. the limit cycle is clearly seen in the noise - free case and the manner in which small noise ,   =  0.2 , and intermediate noise ,   =  0.5 may switch the orbit away from the limit cycle .
when   =  2 , the orbits are close to the noise - free case we further explored the effects of noise on the spiking activity of varying the mean of the input and its noise level in both the current - driven and conductance - driven cases . in fig .  2a , results are shown for the current - driven neuron . the mean number of action potentials ,
n , emitted over a 1,000-ms period , is plotted for various values of the mean current  against the noise level . it should be noted that , in characterizing most central nervous system activity , except for example some brain stem neurons , 1,000  ms is a very long time , as the natural scale is milliseconds . for each data point with noise , 200 trials were performed . 

the mean number of spikes n over a 1,000  ms period is shown as the noise intensity increases for three values of the mean input current . a minimum in the output is clearly seen as  increases when  is near the critical value c .
the mean number of spikes n over a 1,000-ms period is shown as the noise intensity increases for three values of the mean input signal strength ge ( ms / cm ) .
a minimum in the output is clearly seen as e increases when ge is near the critical value at 0.112 and also when ge  =  0.1318  ms / cm a inverse stochastic resonance in the  current - driven 
the mean number of spikes n over a 1,000  ms period is shown as the noise intensity increases for three values of the mean input current . a minimum in the output is clearly seen as  increases when  is near the critical value c .
b inverse stochastic resonance in the  conductance - driven  hodgkin  huxley model .
the mean number of spikes n over a 1,000-ms period is shown as the noise intensity increases for three values of the mean input signal strength ge ( ms / cm ) .
a minimum in the output is clearly seen as e increases when ge is near the critical value at 0.112 and also when ge  =  0.1318  ms / cm without noise (   =  0 ) , there is a critical value of   =  c , which is about 6.44 , at which a stable and unstable limit cycle appear and co - exist with a stable focus . for values of  
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 \begin{document}$$ \mu_{\text{c}}^ { * } $ $ \end{document } with approximate value 9.8 ( hassard 1978 ) , at which a subcritical hopf bifurcation occurs , the system is bistable , and repetitive periodic firing may persist without noise .
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 \begin{document}$$ \mu > \mu_{\text{c}}^ { * } $ $ \end{document } , the focus becomes unstable , and only the stable limit cycle exists ( until a further critical value is reached ) . for values of  between c and \documentclass[12pt]{minimal } 
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 \begin{document}$$ \mu_{\text{c}}^ { * } $ $ \end{document } , perturbations , which may be stochastic or deterministic , can drive the system from near the stable limit cycle or spiking state to near the rest point , or vice - versa . in this article
, we are mainly concerned with the effects of noise for values of  between c and \documentclass[12pt]{minimal } 
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 \begin{document}$$ \mu_{\text{c}}^ { * } $ $ \end{document}. when   =  5.5 , below the critical value c , there is only one spike without noise . as the noise level increases beyond   =  0.2 , so does the mean number of spikes .
however , when   =  6.8 , not far above the critical value and when there are 57 spikes without noise , increasing the noise level makes the mean number n of spikes at first drop dramatically .
there is a pronounced minimum of six spikes at about   =  0.5 , representing a drop in mean spike count of 89% , and thereafter , the mean number of spikes increases at first quite sharply and then more slowly as the noise level increases . for the larger value of the mean input current ,
  =  8 , there are 62 spikes without noise and a noticeable , yet less pronounced , minimum value of 48 in the mean number of spikes when  is just less than one . in order to examine
what role the initial conditions might have played in these results , the values of v , n , m , and h at t  =  0 were all chosen randomly . for v
, the initial value was uniformly distributed from the minimum to the maximum voltage obtained in rhythmic spiking without noise . for n ,
m , and h , their values at t  =  0 were uniformly distributed on ( 0,1 ) .
the effects of additive noise were qualitatively the same as for the case of initial resting conditions .
for example , with   =  6.8 at a time interval of 500  ms , the mean number of spikes dropped from 20.3 without noise to 3.08 as  increased to 0.5 .
thereafter , as  increased , the mean number of spikes increased monotonically to about 31 at   =  4 . with   =  8 , there is a small decline in mean spike count from 29.9 to 24.9 as  increases from 0 to 0.7 , and then a slow increase to n  =  32.4 as  increased further to four .
the results for   =  5.5 also paralleled those for resting initial conditions .
it is pointed out that , with some of the randomly chosen initial conditions , some of the solutions would , in the absence of noise , evolve to the rest point , whereas others would tend to the limit cycle . as a further test of the robustness of the results we have described , we performed simulations in which the initial conditions were fixed at the resting values given in the appendix , but the noise was switched on at a random time tr .
the latter was uniformly distributed over the period of rhythmic spiking , which is about 20  ms .
thus , the neuron is set spiking with a given value of   >  c , and then at tr  =  100  +  20u , where u is a uniform ( 0,1 ) random variable , the noise is switched on .
we recorded the mean number n * of spikes for tr  <  t 
<  500  ms , that is , after the noise is switched on until t  =  500 .
the dependence of n * on  for the two values of  previously employed ( 6.8 and 8) again paralleled those shown in fig . 
for example , with   =  6.8 , the mean spike count n * declined sharply from 21.5 with no noise to a minimum of 4.2 with   =  0.5 .
it may be concluded that the observed inhibitory effect of noise for certain values of the mean input current and the occurrence of a minimum as the noise level increases are robust phenomena for the hodgkin  huxley system . in further support of this latter claim ,
2b shows corresponding results obtained for the case of a conductance - driven neuron , for which the equations are also given in the appendix .
when there is no noise , there is a critical conductance value just less than ge  =  0.112  ms / cm , below which there is no sustained firing .
the bottom curve shows the effects of increasing noise when ge  =  0.0906  ms / cm , which is below the critical value . with increasing noise levels
when ge  =  0.112  ms / cm and there is no noise , there are 55 spikes in a 1,000-ms period .
a small amount of noise causes a very large decrease in spike rate , and as the noise level increases , there is a well - defined minimum at about e  =  0.005 . for a stronger mean stimulus level , ge  =  0.1318  ms / cm ,
there are 63 spikes without noise . as the noise level increases , a distinct minimum occurs at about e  =  0.0075 .
thus , for both kinds of input current , current - driven and conductance - driven , a noise - induced decrease in firing rate occurs for inputs with means near the critical value for periodic spiking , and this inhibitory effect occurs even if the neuron is driven by a purely excitatory synaptic input .
in this article , we reported new and interesting findings on the effects of noise on the behavior of certain dynamical systems relevant to neuronal activity .
these arose from our investigations of the properties of pairs of coupled type 1 ( gutkin and ermentrout 1998 ) neurons with noisy inputs ( gutkin et al .
huxley neurons , which are of type 2 excitability , came from our studies of pairs of such coupled neurons , to be reported elsewhere .
our main observation is that , contrary to what is generally accepted , noise does not always have an excitatory or positive effect ( cope and tuckwell 1979 ; yu and lewis 1989 ; stein et al .
2005 ) , but it can lead to inhibitory or negative effects , which are also amenable to tuning .
the occurrence of a minimum in the response as the noise level increases through a certain value might be called inverse stochastic resonance , a term which derives from its having the opposite character to stochastic resonance , in which a maximum in a response variable occurs as the noise level increases ( collins et al .
huxley variables were chosen randomly or when the noise was switched on at a random time , thus lending support to the robustness of the findings .
preliminary simulations of more complex nerve models and larger networks have yielded the same kind of behavior , both in regard to silencing and the occurrence of a minimum in the firing rate , indicating that these phenomenona are quite general in neural systems .
the effects of noise reported above are explainable in terms of the behavior of the voltage and other variables on what are called stable limit cycles ( murray 1993 ) , which occur when , for example , a neuron fires repetitively at the same frequency ( see fig . 
such a stable limit cycle in a dynamical system often appears by a bifurcation mechanism when a parameter , like the mean input current strength in the hodgkin 
huxley model , varies continuously and crosses some critical value . just above that critical value , the basin of attraction of the limit cycle ,
the stable limit cycle then coexists with one or more other attractors , which is the key to the occurrence of the phenomena we have reported . in the hodgkin 
huxley model , for \documentclass[12pt]{minimal } 
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\mu < \mu_{\text{c}}^ { * } $ $ \end{document } , the only other attractor is a stable quiescent or resting state .
noise can make the solutions leave the basin of attraction of the limit cycle for that of the quiescent state so that spiking ceases .
a minimum in the spike count as noise increases is likely to occur for values of  not far above c .
the transitions to and from the attractors can be explored technically using classical stochastic process theory , as we will report with mathematical detail elsewhere .
when the noise is small , the solution will then typically stay near the rest state for a very long time , but for larger noise , there is a considerable probability that the solutions will get kicked back up to threshold so that spiking may resume .
1b , where the voltage variable v is plotted against the potassium conductance variable n for   =  6.6 , with no noise ,   =  0 , ( top left ) , and values of the noise parameter   =  0.2 , 0.5 , and 2 . it is seen that weak noise can act to switch off the activity and induce a long period without spikes , whereas stronger noise tends to make the system switch back and forth between spiking and non - spiking states .
the rate at which transitions then occur between spiking and non - spiking can be so rapid that the activity seems almost uninterrupted .
the switching behavior for small noise has been observed in recent experiments ( paydarfar et al .
thus , the functional significance of these effects of noise on rhythmic activity is that a very small disturbance can lead to a drastic change in behavior . in the brain ,
electrical activity is often broadly rhythmic , involving limit - cycles in both normal and epileptic activity ( steriade 2000 ; buzski and draguhn 2004 ) .
if such oscillations arise near a bifurcation point , then a small noisy signal could lead to the cessation of , or a sharp modification of , rhythmic activity .
it is feasible that this could be the basis of a therapeutic approach to alleviate symptoms in the case of pathological oscillatory activity .
also , in a population of cells , for example , with functionally tuned receptive fields , those which are weakly responding and operating near a bifurcation point akin to c could be silenced easily by noisy inputs , whereas cells firing at higher levels would have their activity augmented by noise , giving noise - induced tuning to the population responses .
this may represent a new noise - induced mechanism for the sharpening of neural responses .
note that the perturbing inputs ( noise ) do not have to be smooth but could be just as effective if they had an impulsive nature . since dynamical systems in diverse fields exhibit stable limit cycles coexisting with a stable focus , we expect to find that the phenomena of inhibition of cyclic , repetitive , or rhythmic activity by noise and inverse stochastic resonance will have widespread occurrence .
examples of fields or systems where a stable limit cycle coexists with a stable rest state occur in circadian rhythms ( jewett and kronauer 1998 ) , cardiology ( panfilov and holden 1997 ) , cell kinetics and tumor growth ( lahav et al .
2004 ; goldbeter 1991 ) , and oscillating neural networks ( steriade 2000 ; buzski and draguhn 2004 ) , as well as in climatology , ecology , and astrophysics .
the underlying mathematical bifurcation pattern suggests that the phenomena we have detected are of a general nature and not restricted to the hodgkin 
although the phenomena we have described are of interest in themselves , as indeed is stochastic resonance , their functional significance in neurobiological and other dynamical systems remains to be fully explored . a related finding
was reported in a heuristic nonlinear stochastic model of affective disorders ( huber et al . , 2004 ) .
the original hodgkin  huxley system ( hodgkin and huxley 1952 ) is employed as follows , in which v(t ) is the depolarization from resting potential in millivolts at time t ( ms ) and n(t ) , m(t ) , and h(t ) are the ( dimensionless ) auxiliary variables for potassium activation , sodium activation , and sodium inactivation , respectively : 
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$ $ \end{document}here , c is the membrane capacitance per unit area ;  , which may depend on t , is the mean input current density ; and \documentclass[12pt]{minimal } 
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the input current has a stochastic component , w being a standard wiener process and  being the noise amplitude .
the voltage - dependent coefficients in the auxiliary equations are 
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for conductance - based noise , the current terms dt  +  dw in the voltage equation are replaced by \documentclass[12pt]{minimal } 
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 \begin{document}$$ g_{\text{i}}^ { * } $ $ \end{document } are equilibrium values , e and i are time constants , we and wi are independent standard wiener processes , and e and e are constant noise amplitudes . the parameters employed in the simulations , with excitation only , were ve  =  80  mv , e  =  2  ms .
 the effects of noise on neuronal dynamical systems are of much current interest . here 
 , we investigate noise - induced changes in the rhythmic firing activity of single hodgkin  
 huxley neurons . with additive input current 
 , there is , in the absence of noise , a critical mean value   =  c above which sustained periodic firing occurs . with initial conditions as resting values , for a range of values of the mean  near the critical value 
 , we have found that the firing rate is greatly reduced by noise , even of quite small amplitudes . 
 furthermore , the firing rate may undergo a pronounced minimum as the noise increases . 
 this behavior has the opposite character to stochastic resonance and coherence resonance . 
 we found that these phenomena occurred even when the initial conditions were chosen randomly or when the noise was switched on at a random time , indicating the robustness of the results . 
 we also examined the effects of conductance - based noise on hodgkin  
 huxley neurons and obtained similar results , leading to the conclusion that the phenomena occur across a wide range of neuronal dynamical systems . 
 further , these phenomena will occur in diverse applications where a stable limit cycle coexists with a stable focus . 
Introduction
Methods
Results
Discussion
A. HodgkinHuxley equations with additive noise
B. HodgkinHuxley equations with conductance-based noise