Detection of subthreshold pulses in neurons with channel noise
aa r X i v : . [ q - b i o . N C ] N ov Physcial Review E , 051909 (2008). Detection of subthreshold pulses in neurons with channel noise
Yong Chen,
1, 2, ∗ Lianchun Yu, and Shao-Meng Qin Institute of Theoretical Physics, Lanzhou University, Lanzhou , China Key Laboratory for Magnetism and Magnetic materials of the Ministry of Education, Lanzhou University, Lanzhou , China (Dated: October 25, 2018)Neurons are subject to various kinds of noise. In addition to synaptic noise, the stochastic opening and closingof ion channels represents an intrinsic source of noise that a ff ects the signal processing properties of the neuron.In this paper, we studied the response of a stochastic Hodgkin-Huxley neuron to transient input subthresholdpulses. It was found that the average response time decreases but variance increases as the amplitude of channelnoise increases. In the case of single pulse detection, we show that channel noise enables one neuron to detectthe subthreshold signals and an optimal membrane area (or channel noise intensity) exists for a single neuron toachieve optimal performance. However, the detection ability of a single neuron is limited by large errors. Here,we test a simple neuronal network that can enhance the pulse detecting abilities of neurons and find dozensof neurons can perfectly detect subthreshold pulses. The phenomenon of intrinsic stochastic resonance is alsofound both at the level of single neurons and at the level of networks. At the network level, the detection abilityof networks can be optimized for the number of neurons comprising the network. PACS numbers: 87.19.lc, 87.19.ln, 87.16.Vy, 87.19.lb, 05.40.-a, 07.05.Tp
I. INTRODUCTION
It is well known that neurons are subject to various kinds of noise. Intracellular recordings of cortical neurons in vivo consis-tently display highly complex and irregular activity [1], resulting from an intense and sustained discharge of presynaptic neuronsin the cortical network. Previous studies have suggested that this tremendous synaptic activity, or synaptic noise, may play aprominent role in neural information transmission as well as in neural information processing [2]. For example, with stochasticresonance (SR), synaptic noise facilitates information transfer or allows the transmission of the subthreshold inputs [3]. Indeed,SR induced by synaptic noise has been extensively studied in a single neuron and neural populations both experimentally andnumerically [4, 5, 6].While the synaptic noise accounts for the majority of noise in neural systems, another significant noise source is the stochasticactivity of ion channels. Voltage-gated ion channels in neuronal membranes fluctuate randomly between di ff erent conformationalstates due to thermal agitation. Fluctuations between conducting and non-conducting states give rise to noisy membrane currentsand subthreshold voltage fluctuations. Recently, much e ff ort has been devoted to this field and channel noise is now understoodto have important e ff ects on neuronal information processing capabilities. Studies show that channel noise alters action potentialdynamics, enhances signal detection, alters spike-timing reliability, and a ff ects the tuning properties of the cell [7, 8, 9, 10] (forreview see [11]).Detection of small signals is particularly important for animal survival [13]. Both experimental and numerical studies havefound, as depicted by SR, that synaptic noise can enhance the detection of subthreshold signals in nonlinear and threshold-detecting systems. For channel noise, there have been many papers concentrating on SR induced by channel noise [14, 15], andtheir results suggest that neurons may utilize channel noise to process subthreshold signals. However, it is still unclear whetherreliable detection of subthreshold signals could obtained for single neuron if neurons do utilize SR to process signals. On theother hand, as a intrinsic noise source of neurons, channel noise is mostly studied within single neurons. Since recent studiessuggest that channel noise enhances synchronization of two coupled neurons[16], it is natural to ask whether channel noise couldtake e ff ects in the network level.In this study we focus on subthreshold pulse detection in neurons with channel noise. First, using the stochastic Hodgkin-Huxley (SHH) neuron model, we study the e ff ects of channel noise on the response properties of a single neuron to subthresholdpulse input. We find that a SHH neuron fires spikes a higher than average level in response to a subthreshold stimulus. Theaverage response time decreases while the variance increases as the channel noise amplitude increases. This result is explainedwell by the phase plane analysis method [17]. Then, we evaluate the subthreshold signal detection ability of a SHH neuronunder the pulse detection scenario proposed by Wenning et al. [18]. They reported that colored synaptic noise can enhancethe detection of a subthreshold input. However, since the total error is always greater than 0 .
5, they argued that biologicalrelevance of pulse detection for a single neuron is questionable. In the case of channel noise, we come to a similar conclusion. ∗ Corresponding author. Email: [email protected]
Therefore, we propose a feasible solution for a neuronal population to overcome this predicament. We find that subthresholdsignal detection can be greatly enhanced with the neuronal networks we propose. The phenomenon of intrinsic SR induced bychannel noise is also observed. We argue this SR may be a strategy that neural systems would take to optimize their detectionability for subthreshold signals.Our paper is organized as follows. In Sec. II the stochastic version of the Hodgkin-Huxley neuron model is presented. In Sec.III, we focus on how the single neuron responses to subthreshold transient input pulse. Phase plane analysis method is appliedto explain results presented. In Sec. IV, we present the simple scenario for pulse detection and demonstrate that the detectionability of a single neuron is limited. Then we introduce the network that could reliably detect subthreshold pulses. Discussionsand conclusions are presented in Sec. V.
II. MODELSA. Deterministic Hodgkin-Huxley Model
The conductance-based Hodgkin-Huxley (HH) neuron model provides a direct relationship between the microscopic proper-ties of an ion channel and the macroscopic behaviors of a nerve membrane [19]. The membrane dynamics of the HH equationsare given by C m dVdt = − (cid:16) G K ( V − V revK ) + G Na ( V − V revNa ) + G L ( V − V L ) (cid:17) + I , (1)where V is the membrane potential. V revK and V revNa , V L are the reversal potentials of ( K ) potassium and ( Na ) sodium, theleakage currents, respectively. G K , G Na , and G L are the corresponding specific ion conductances. C m is the specific membranecapacitance, and I is the current injected into this membrane patch. The conductance for potassium and sodium ion channels aregiven by G K ( V , t ) = g K n , G Na ( V , t ) = g Na m h , (2)where g K and g Na are products of two factors: an individual channel conductance γ K and γ Na respectively, and the channeldensities ρ K and ρ Na respectively. g K and g Na give the maximum conductance when all channels are open. The gating variables, n , m , and h , obey the following equations, ddt n = α n ( V )(1 − n ) − β n ( V ) n , ddt m = α m ( V )(1 − m ) − β m ( V ) m , ddt h = α h ( V )(1 − h ) − β h ( V ) h , (3)where α x ( V ) and β x ( V ) ( x = n , m , h ) are voltage-dependent opening and closing rates and are given in Table I with the otherparameters used in the following simulations. TABLE I: Parameters and rate functions used in our Simulations. C m Specific membrane capacitance 1 µ F / cm V revK Potassium reversal potential − mVV revNa Sodium reversal potential 50 mVV L Leakage reversal potential − . mV γ K Potassium channel conductance 20 pS γ Na Sodium channel conductance 20 pSG L Leakage conductance 0 . mS / cm ρ K Potassium channel density 20 /µ m ρ Na Sodium channel density 60 /µ m α n . V + − e − ( V + / ms − β n . e − ( V + / ms − α m . V + − e − ( V + / ms − β m e − ( V + / ms − α h . e − ( V + / ms − β h + e − ( V + / ms − B. Stochastic Hodgkin-Huxley Model
The deterministic HH model describes the average behaviors of a larger number of ion channels. However, ion channels arerandom devices, and for the limited number of channels, statistical fluctuations play a role in neuronal dynamics [20]. To treatthe consequent fluctuations in ion conductance, two kinds of methods are often employed.One is the so-called Langevin method which characterizes channel noise with Gaussian white noise [21]. In this description,the voltage variables still obey Eqs. (1) and (2) but the gating variables are random quantities obeying the following stochasticdi ff erential equations, ddt n = α n ( V )(1 − n ) − β n ( V ) n + ξ n ( t ) , ddt m = α m ( V )(1 − m ) − β m ( V ) m + ξ m ( t ) , ddt h = α h ( V )(1 − h ) − β h ( V ) h + ξ h ( t ) , (4)where the variables ξ n ( t ), ξ m ( t ), and ξ h ( t ) denote Gaussian zero-mean white noise with (cid:10) ξ n ( t ) ξ n ( t ′ ) (cid:11) = N K α n ( V )(1 − n ) − β n ( V ) n δ ( t − t ′ ) , (cid:10) ξ m ( t ) ξ m ( t ′ ) (cid:11) = N Na α m ( V )(1 − m ) − β m ( V ) m δ ( t − t ′ ) , (cid:10) ξ h ( t ) ξ h ( t ′ ) (cid:11) = N Na α h ( V )(1 − h ) − β h ( V ) h δ ( t − t ′ ) , (5)where N K and N Na are the total number of K + and Na + channels. Note that in this description, a precondition is that n , m , and h should be in the interval [0 , ff ective method and widely used for its low computational cost. Additionally, the trajectory of thephase point prior to a spike entails major changes in the variables V and m but the variables n and h are practically unchangedduring the same epoch [23]. So, this recipe enables us to investigate system behaviors in the V − m phase plane. (b) h h [m h ] m [m h ] m [m h ] m m [m h ] mm [m h ] m [m h ] m [m h ] m m [m h ] mmh h h h h h [n ] n [n ] n [n ] n n [n ] n n [n ] nn (a) FIG. 1: Kinetic scheme for a stochastic potassium channel (a) and sodium channel (b). n and m h are open states, while the other states arenon-conducting. Another method is based on the assumption that the opening and closing of each gate of the channel is a Markov process. Withthis methods, the ion channel stochasticity is introduced by replacing the stochastic equations by the explicit voltage-dependentMarkovian kinetic models for a single ion channel [8, 11, 12]. As shown in Fig. 1, the K + channels can exist in five di ff erentstates and switch between these states according to the voltage dependence of the transition rates (identical to the original HHrate functions). n labels the single open state of the K + channel. The Na + channel kinetic model has 8 states, with only oneopen state m h . Thus the voltage-dependent conductances for K + and Na + channels are given by G K ( V , t ) = γ K [ n ] / S , G Na ( V , t ) = γ Na [ m h ] / S , (6)where γ K and γ Na are defined as before, and [ n ] refers to the number of open K + channels, [ m h ] the number of open Na + channels, and S the membrane area of the neuron.The numbers of open K + and Na + channels at a special time t is determined by the following formula: if the transition ratebetween state A and state B is r and the number of channels in these states is denoted by n A and n B , the probability that a channelswitches within the time interval ( t , t + ∆ t ) from state A to B is given by p = r ∆ t . Hence, for each time step, we determine ∆ n AB ,the number of channels that switch from A to B , by choosing a random number from the following binomial distribution, P ( ∆ n AB ) = n A ∆ n AB ! p ∆ n AB (1 − p ) ( n A − ∆ n AB ) . (7)Then we update n A with n A − ∆ n AB , and n B with n B + ∆ n AB . To ensure that the number of channels in each state is positive,starting at the beginning with the largest rate, we update these numbers sequentially, and so forth [22].The noisiness of a cluster of channels can be quantified by the coe ffi cient of variation (CV) of the membrane current. Underassumptions of stationarity ( V is fixed ), CV = { (1 − p ) / np } / , where n is the number of channels and p is the probabilityfor each channel to be open. Thus the noisiness for a given population of voltage-gated channels is proportional to n / [11].Accordingly, in this study, we introduce the membrane area S as a control parameter of the channel noise level. Given ionchannel density, the level of channel noise decreases with an increase in membrane area.The numerical integrations of stochastic equations for both the occupation number method and the Langevin method areperformed by using forward Euler integration with a step size 0 . ms . The parameters used in all simulations are listed inTable I. The occurrences of action potentials are determined by upward crossings of the membrane potential at a certain detectionthreshold 10 mV if it has previously crossed the reset value of − mV from below. III. THE RESPONSE OF SHH NEURON TO A SUBTHRESHOLD TRANSIENT INPUT PULSE
The signal detection of transient subthreshold input pulses has received increasing attention in recent years [24, 25, 26] (see[18] for more references). In our study of the response of a SHH neuron, the transient input pulses are set with width δ t = . ms and strength I = µ A / cm .Fig. 2(a) depicts the post-stimulus time histograms (PSTHs) of a SHH neuron with a membrane area S =
20, 200, and1000 µ m , respectively. Each stimulus was repeated 5000 times. The number of spikes observed in each bin (bin size = . ms )is normalized by the total number of stimuli and by the bin size. Thus, the PSTH gives the firing rate or the distribution of thefiring probability as a function of time [27]. Obviously, there exists a peak over the spontaneous firing level in each curve andthe peak lessens as the membrane area S increases. The higher the peak, the more sensitive neuron responses are to stimuli,which are activated by channel noise. The baselines show the average level of spontaneous firing due to channel noise. With ahigher baseline, the number of spontaneous spikes increases. R. K. Adair has shown that the firing rate of a neuron with channelnoise can be reduced by lowering the resting potential (Fig. 5 in Ref. [14]). In our case, the transient input pulse temporallyhold the resting potential to a high state, thus gives a temporally higher firing rate over the spontaneous one. As the membranearea increases, since the fluctuations in membrane currents become smaller, the firings in response to the subthreshold signalsas well as the noise-induced spontaneous firings are reduced, yielding reductions in heights of both the peaks and the baselines.It is noted that adjacent to the peak, there follows a time interval of about 10 ms during which the firing rate is below its averagelevel. We argue that this trough shape of the PSTH is due to refractoriness of the neurons [19]. If in a certain time interval thefiring rate is higher than its average level, the firing rate in the following time range will be reduced because refractory e ff ectprevents occurrence of the immediately following firings. The time interval of 10 ms is in accordance with e ff ective refractoryperiod reported by other researchers [28].To find the range in the membrane area which is more sensitive to a pulse than channel noise perturbation, we define signal-to-noise ratio (SNR) as the ratio of increased firing probability in response to input pulses to the probability for spontaneous firingin response to channel noise. [27]. As shown in Fig. 2(b), when the membrane area is smaller than 100 µ m , SNR remains verysmall. With increasing membrane area, SNR increases rapidly and reaches its maximum at about 400 µ m . However, furtherincreasing the membrane area leads to a decreasing in SNR. This figure clearly demonstrates the phenomenon of stochasticresonance. It is noted that as the membrane area increases, both the peak and the baseline of PSTH is reduced, so the occurrenceof SR for SNR curve is a result of trade-o ff between neuron’s sensitivity to subthreshold signals and rejection of spontaneousfirings.Next, we investigate how the channel noise a ff ects the response time of neurons to subthreshold signals. It has been recentlyproposed that the first spikes which occur in, for example, cortical neurons, may contain information about a stimulus [29]. Thus,determinacy in response time of a neuron to signals is relevant to the information content, and how it is a ff ected by channel noisewould be an important question to explore [23]. The PSTH analysis provides us a first glimpse into it. The central positions of thePSTH peaks represent the mean response time, and the widths of the peaks represents the variances in response time. We see thatas membrane area is increased, the central position of PSTH peak moves rightward, and its width is reduced simultaneously[seeFig. 2(a)]. This implies that as the membrane area increases, the mean response time increases but the variance in response time S p i k e R a t e ( H z ) time (ms)
20 m
200 m (a) S N R membrane area ( m ) (b) < T > Var ( T )membrane area ( m ) < T > ( m s ) V a r ( T ) (c) FIG. 2: (Color online) The response properties of a SHH neuron to subthreshold transient input pulses. (a) The PSTH of a SHH neuron withdi ff erent membrane areas vs time. (b) SNR vs membrane area. (c) the mean response time and the variance of response time vs membranearea. decreases. This is in consistent with the results obtained in the case of subthreshold inputs for external noise [30]. From 5000times repeated trials, we directly calculate the mean and variance of response time for di ff erent membrane area, and plot themin Fig. 2(c). It seems that the increasing of h T i as well as the decreasing of Var ( T ) is nearly in an exponential form.To investigate the dynamic mechanism of a SHH neuron responding to transient input pulses, we performed a phase planeanalysis with the Langevin simulation model described above. δ t and I for the input pulses are set as 1 ms and 6 µ A / cm ,respectively. Fig. 3(a) shows the stable fixed point (SFP), part of the action potential trajectory (APT), and the unstable circles(UC) corresponding to di ff erent intensities of the input pulses in a noise-free HH model. The whole APT for I = . ff ect the length of the pulsedisplacement, and so the jumping area is determined by the initial state of the systems before the input pulse is applied (in ourcase, the state is described by two variables: V and m ; n and h are not considered). The initial states with larger V and m aremore likely to lead to an action potential [see the averaged initial positions for firing and no firing plotted in Fig. 3(c)]. Thereforewe conclude that the response of the single SHH neuron to input pulses is state-dependent. FIG. 3: (Color online) The phase plane analysis for the response properties of SHH neuron with Langevin simulations. (a) The action potentialtrajectory (APT) and the unstable circle of a noise-free HH neuron in the phase plane. Inset: the overall APT for I = . S = µ m . (c) The noise-induced APT and the unstable circle (UC) for S = µ m . (d) Equitime labeling analysis of the response time to di ff erent initial states. We also investigated the temporal response of the SHH neuron in the V − m phase plane. In Fig. 3(d), the APTs with threedi ff erent response times are traced and labeled with bars separated equally by 0 . ms . The leftmost bars denote the time thatthe input pulse is applied, and the dashed line denotes the time that the spikes are detected. It shows that the system reaches aposition closer to the detection threshold if the initial state is higher, and it will come into the outer side of the APT on which thesystem moves more quickly than the systems on the inside of the APT. As a result, this system presents a shorter response timeto the input pulse, and vice versa. We see that the response time for a particular input pulse is dependent on the initial state ofthe system. In addition, one could deduce that it is the variance of initial state that results in the variance of the response time.Next, we investigated how the change of membrane area (ie., the channel noise level) e ff ects the distribution of initial state ofthe system, so that the response time exhibits statistics as shown in (c) of Fig. 2. The distributions of initial state for di ff erentmembrane areas are described by the average and variance of V ini and m ini , which are calculated from 2000 firings in response tothe pulses with Langevin simulation. It is seen from Fig. 4 that as the membrane area increases, both the average and variance of V ini and m ini decreases. In other words, with the decreasing of channel noise, the distribution of initial states in the phase planemoves left-down to the lower value and becomes narrower. As we discussed above, lower initial state leads to longer responsetime, and narrower distribution of initial states leads to smaller variance of the response time. Therefore, the average responsetime is prolonged and its variance is reduced if the membrane area is increased.It is noted that because of large computational cost of our model, it is di ffi cult to obtain the statistical properties of responsetime for larger membrane area. But through the phase plane analysis we see that the most inner APT results in the maximalresponse time, which is the up limit of the average response time. As the membrane area increases to infinitely large, the average < m i n i >
20, 50, 80, 100, 200, 500, 800, 1000 µ m , respectively. M M CC MMMF V ( m V ) t (ms) F M C
FIG. 5: Trace of the membrane potential of the SHH neuron for transient input pulses with width of 0 . ms and height of 5 µ A / cm . The timeof occurrence for pulses is marked by arrowheads on the horizontal axis. C : the incidences that the pulse is correctly detected by the neuron; M : the incidences that the neuron does not respond to the pulse; F : the incidences that a spike occurs in the absence of a pulse. response time will increase gradually to this maximal value. We argue this maximal response time corresponds to the responsetime of the noise-free HH neuron to the pulse of which the strength to elicit a spike is minimal. In the deterministic HH model,for the pulse input with the minimal strength of I = . µ A / cm and δ t = . ms , the maximal response time we obtainedis 5 . ms , which matches the PSTH peaks’ right edges (see curves for 200 µ m and 1000 µ m in Fig. 2(a)). This time scale isimportant for the choosing of the coincidence time window ρ introduced in section IV. If ρ is much larger than the maximalresponse time the SHH neuron could provide, spontaneous firings will be detected together with the stimulated firings, thusreduce the accuracy of detection. On the contrary, if ρ is far less than it, some stimulated firings will be ignored, so the e ffi ciencyof detection is reduced. IV. THE PERFORMANCE OF PULSE DETECTION
Now, we consider the pulse detection task as a simple computation that a neural system can perform, in which we evaluate theperformance of a single SHH neuron as well as a SHH neuron assembly.The input I stim is modeled as a serial narrow rectangular current pulse with width δ t = . ms and strength I = µ A / cm (see Fig. 5). The input pulse train (the arrowheads on the horizontal axis) is regular with a large time interval ∆ T = ms .Compared to the membrane time constant, the preceding pulse has no significant influence on the following one. In such anarrangement, as has been discussed above, the SHH neuron has three di ff erent responses (marked with di ff erent capital lettersrespectively in Fig. 5) to the pulse train which consists of n equidistant pulses:(1) C : The neuron generates an action potential immediately (within the time rang of 5 ms ) after a pulse is presented, whichsignifies successful detection of the pulse. We define P C as the fraction of correctly detected pulses, which is the totalnumber of correctly detected pulses, divided by the total number of input pulses.(2) M : The neuron fails to fire a spike immediately (within the time rang of 5 ms ) after the pulse is presented. If we define P M as the fraction of missed pulses, then we have P M = − P C .(3) F : The neuron fires a spike in the absence of an input pulse (a false positive event). A deterministic HH neuron cannotfire spikes when the stimulus is not applied or if it is below the threshold. However, in the case of channel noise, stochastice ff ects give rise to spontaneous spiking. To describe the e ff ect of those spontaneous firing spikes on subthreshold pulsedetection, we denote P F as the total number of false positive events divided by the total number of input pulses. Note that P F can easily exceed 1.In order to quantify the neuron’s response to the pulse train, we define the total error Q for the pulse detection, Q = P M + P F . (8)For a longer interval ∆ T with fixed n , the false positive events are more likely to occur and the total error grows with increasing ∆ T .Fig. 6(a) shows P C , P M , and P F as a function of the membrane area under the above-mentioned scenario. According to theabove PSTH analysis , when the membrane area is rather large, though the system could be displaced by subthreshold pulsesto near the threshold, the channel noise is small and can hardly trigger firings, thus P C is very small and P M ≈
1. The firingstriggered by noise alone is even less, so the P F is smaller than P C . When the membrane area is small, the channel noise isremarkable, giving large P C . Meanwhile, due to the high rate of spontaneous firings, P F is even larger than P C . So, withincreasing membrane area, both P C and P F decreases, but P F drops more quickly than P C . When the membrane area is largerthan about 180 µ m , P C becomes larger than P F .The total error Q as a function of the membrane area is also plotted in Fig. 6(b). As the membrane area increases, due to therapid decline of P F , the total error Q drops rapidly. Then, with further increase in membrane area, Q increases and approaches 1for the major contribution from the fraction of missed pulses. Because Q is basically the summation of ascending P M curve anddescending P F curve, one can expect a minimal value for it. The minimal value of Q is 0 . S = µ m . With this optimalmembrane area, we see that the neuron achieves balance between detecting pulse input and suppressing spontaneous firings.It should be noted that the positions for P C , P M and P F curves are dependent on the strength of pulse input I , or the interpulseinterval ∆ T . Changing the pulse strength would change the position of P C curve, thus the position of P M in Fig. 6(a). Inparticular, smaller pulse strength leads to lower position of P C , thus higher position of P M (see Fig. 7 of Ref.[18] or Fig. 6 ofRef.[14]). As discussed above, the pulse induced high firing rate would reduce the spontaneous firing rate through refractorinesswithin the following 10 ms . If ∆ T is large compared to refractory period of SHH neuron, this reduction in spontaneous firings isnegligible, so the pulse strength would not a ff ect the position of P F curve. On the contrary, by their definition, P F , rather than P C and P M , is greatly dependent on ∆ T . Whatsoever, neither pulse strength nor interpulse interval will not change the overall shapeof both P C and P F curves. Since the minimal Q is basically the result of summation of ascending P M curve and descending P F curve, we argue that there is always a minimal value for Q , and the optimal membrane area for Q di ff ers for di ff erent input pulsestrength or interpulse interval.We see that the single neuron has limited capacity for subthreshold signal detection. The channel noise is basically a zero-meannoise, which means the probability for a subthreshold pulse gets enhanced by a positive fluctuation is equal to the probabilitythat it is further suppressed by its negative counterpart. In more detail, the response of the SHH neuron to input current pulses isstate-dependent(Fig. 3(c)). Channel noise perturbations enable the system, with equal chance, to be in a high state that the neuronis more likely to fire a spike after pulse is applied, or in a low state that no fires occur. As a result, P C could never exceed 0.5, andthe total error for a single neuron is always larger than 0.5. Indeed, we found in Fig.4 of Ref. [14] that the spike e ffi ciency forsubthreshold voltage impulses never exceed 50% and the same conclusion was made in Ref. [18] for external noise. So we seethat theoretically, it is unlikely to utilize channel noise to reliably detect subthreshold signals with single neuron. However, inreality, the neuron assembly works in real neural systems rather than in a single neuron. In general, neurons work cooperativelythrough synaptic coupling. What’s more, among various spatiotemporal spike patterns in the neural system, synchronous firinghas been most extensively studied both experimentally and theoretically [31, 32]. Believing that neuronal synchronous firingis critical for transmitting sensory information, many investigators have suggested that a major function of cortical neurons isto detect coincident events among their presynaptic inputs (see [33] for more references). Based on this fact, we proposed aneuronal network that can greatly enhance the detection ability of the pulse. P C , P M & P F membrane area ( m ) P C P M P F (a) Q membrane area ( m ) FIG. 6: (Corlor online) The performance of a single neuron in a subthreshold pulse detection task. (a) P C , P M , P F , and (b) total error Q as afunction of membrane area. As shown in Fig. 7, the front layer of the network is composed of globally coupled identical neurons with channel noise. Thecoupling term has the form of an additional current I couple added to the equation for the membrane potential (see Eq. 1). For the i th neuron, it takes the form I couple = ε N N X j = ( V j − V i ) , (9)where ε is the coupling strength and V i is the membrane potential of the i th neuron for i = , . . . , N . In our simulations, neuronsare weakly coupled, ε = . S = µ m . Here we chose this valuefor the membrane area rather than the optimal one so that P C of a single neuron is relatively large. Thus fewer neurons areneeded in our network and the computational cost is consequently reduced. Each SHH neuron in the network receives the samesubthreshold pulse train as in the single neuron case. The output spike trains of those neurons S i ( i = , , . . . , N ) are takenas the input of a so-called coincidence detector (CD) neuron. In neural reality, coincidence detection requires complex cellularmechanisms [34, 35]. For simplicity, here we use a phenomenological CD neuron model. The CD neuron is excited when itdetects spikes from more than θ neurons within a coincidence time window ρ ( = ms , see discussion Sec. V). In other words, θ denotes the detection threshold of the CD neuron. After firing, the CD neuron enters a refractory period of 5 ms . Obviously,given the input spike trains S i , the output spike train R θ of the CD neuron is determined by its threshold θ . We also define P θ C and P θ F as the fraction of correct detection and false reporting in the network with the CD threshold θ , respectively. Similarly,0 s N s SHHSHHSHH CD I stim s R FIG. 7: A schematic diagram of pulse detection with multiple neurons. I stim is the input pulse train. S i is the output spike train of the i th SHHneuron for i = , , . . . , N . CD is the coincidence detector neuron and R θ is its output spike train with synchronous firing detecting threshold θ . Here we demonstrate the case of θ = Q θ is defined as the total error of the network with the CD threshold θ . TABLE II: The performance of the network with optimal sizes for di ff erent CD threshold θ . θ N opt ∼
65 69 ∼
91 78 ∼ Q min P θ C Fig. 8(a) shows P θ C as a function of the number of SHH neurons N for θ = , ,
3. As the number of neurons N increases,all P θ C increase quickly to P θ C =
1. For larger θ , the increase of P θ C becomes slower and requires more neurons to achievethe successful state P θ C =
1. However, the enhancement of P θ C is at the cost of unexpected improvement in P θ F . As shown inFig. 8(b), with increasing N , P θ F is also improved. Note that P θ F is able to exceed 1. Comparing Fig. 8(a) with (b), it is obviousthat, though both P θ C and P θ F increase with an increased number of neurons, comparing to P θ C , the increasing of P θ F is alwaysdelayed. Thus, in the case of a small N , the correct detection will not be greatly enhanced though P θ F is low. Whereas for large N , one can obtain better performance for correct detection but a cost of a higher P θ F . Therefore, we expect to find an optimal N to achieve the best performances for signal detection.Fig. 8 (c) displays the total error Q θ as a function of the number of neurons N for di ff erent θ . Clearly, the minimal totalerror Q min or the resonance behavior appears at the network level. For di ff erent θ , there exist di ff erent optimal numbers of SHHneurons N opt where the performance of pulse detection is at its best. As shown in Table II, with increasing θ , Q min decreaseswhile the corresponding N opt increases. Simultaneously, P θ C corresponding to the N opt increases. If θ is large enough, the Q min becomes nearly zero ( < . Q min could appear in a wider range of SHH neuron numbers.We define syn-firing probability P θ as the probability that θ or more than θ SHH neurons fire in a time interval. Supposing thefiring probability of each independent SHH neuron (ignoring the couplings between them) in a time interval is p , then syn-firingprobability P θ in this time interval is described by cumulative distribution function for a binomial distribution, i.e., P θ = N X α = θ N ! α !( N − α )! p α (1 − p ) N − α , (10)where p α (1 − p ) N − α is the probability that only α neurons fire at the same time, and C N α = N ! α !( N − α )! is the number of ways ofpicking α neurons from population N . So P N α = θ N ! α !( N − α )! is the total number of ways of selecting θ or more than θ neuron out ofpopulation N . Then the firing probability for CD neuron with threshold θ and refractoriness is written as P θ CD = P θ − ( in f luence o f past perturbations ) , (11)where the last term represents the suppression of past firings on present firing probability through refractoriness, which is propor-tional to the probability for past firing events [36]. This term can be cancelled because it often acts as small perturbations, and1 P C number of SHH neurons = 1 = 2 = 3 = 4 = 5 (a) (b) P F number of SHH neurons = 1 = 2 = 3 = 4 = 510 (c) Q number of SHH neurons = 1 = 2 = 3 = 4 = 5 FIG. 8: (Color online) Detection of subthreshold signals with the neuronal network for the CD threshold θ = , , ...,
5. (a) P θ C , (b) P θ F , and (c) Q θ as a function of number of SHH neurons. does not bring qualitative changes to firing probabilities of CD neuron statistically. So in the following analysis, for simplicity,we take P θ CD ≈ P θ .From Fig. 9(a) we see that P θ , thus P θ CD increases with increasing p , the firing probability of each neuron. So when pulsesare applied to the SHH neuron, the firing probability of CD neuron is larger than that caused by channel noise alone, because thefiring probability of each SHH neuron is enhanced. By increasing the number of SHH neurons, no matter p is large or small, P θ ,thus P θ CD increases(Fig. 9(b)). Since as in the single neuron case, P θ C is proportional to P θ CD in response to subthreshold pulses,and P θ F is proportional to P θ CD induced by channel noise alone. P θ C and P θ F rise, but P θ M declines (not shown) as the number ofSHH neurons increases. Therefore, as a result of summation of declining P θ M curve and ascending P θ F curve, a minimum of thetotal error Q θ is warranted.As shown in Fig. 9(c), with the increasing of θ , P θ drops, and for small p , P θ drops more quickly. So both P θ C and P θ F curvesmove right-down in Fig. 8 (a) and (b), and P θ M curves move right-up(not shown). Thus, Q min curves would move rightward inFig. 8(c). Since for single SHH neuron, its firing probability is low when pulse inputs are absent, P θ F curves move more rapidthan the P θ M curves. As a result, the Q θ curves become wider, and move also downward in Fig. 8(c) as θ increases. So we seethat the drop of Q min is warranted and Q min = θ is largeenough. V. DISCUSSION AND CONCLUSION
In this paper, we used the stochastic version of Hodgkin-Huxley neuron model in which channel noise is the only source ofnoise, and discussed the possibility of detecting subthreshold signals with channel noise.First, we studied the response property of the single SHH neuron to the subthreshold transient input pulses. The main result2 -4 -3 -2 -1 P p (a) (b) P number of SHH neuron p=0.1 p=0.05 p=0.01 p=0.005 p=0.0012 4 6 8 10 12 1410 -30 -25 -20 -15 -10 -5 (c) P p=0.1 p=0.05 p=0.01 p=0.005 p=0.001 FIG. 9: (Corlor online) The syn-firing probability P θ of the network calculated from Eq. 10. (a) P θ as a function of p , the firing probabilityof each SHH neurons. θ = N = P θ as a function of number of SHH neurons for di ff erent p . θ =
3. (c) P θ as a function of θ fordi ff erent firing probability of each SHH neurons p . is that the SHH neuron fires spikes with a higher rate over its average level in response to a subthreshold stimulus. The averageresponse time decreases but its variance increases as the channel noise amplitude increases (or with decreasing membrane area).We further found the existence of an up limit for the average response time. From phase plane analysis we see that this uplimit should be predictable for threshold systems with any zero-mean noise, as the noisiness decreases. This results means theresponse time is very sensitive to the membrane area, because a small decreasing of membrane area would lead to remarkabledecreasing in mean response time and increasing in its variance.Adair has demonstrated the stochastic resonance in ion channels as the output response (in the probability of action potentialspikes, which is equivalent to P C in our paper.) from small input potential pulses across the cell membrane is increased by addednoise, but falls o ff when the input noise becomes large. However, to evaluate the reliability of subthreshold signal detection,one must consider not only the response to subthreshold signals but also the spontaneous firings, because from the standpointof a neuron, those two kinds of output make no di ff erence. In this paper, we endowed the SHH with a simple pulse detectionscenario and calculated the total error Q . We found that a minimal Q and the corresponding optimal membrane area (noise). Sowe argue that to maximize the detection ability, the strategy a neuron should take is balancing between response to pulses andrejecting spontaneous firings, rather than improving the response to pulses alone with optimal noise as Adair demonstrated. Aswe argued, the first strategy allow to achieve the minimal Q for di ff erent pulse strengths, unlike the second one with which is ine ff ect only for large pulse strength (see Fig. 6 in Ref. [14]). However, even with the first strategy, we found the detection abilityof a single neuron is non-credible because Q cannot be larger than 0.5. Though the results are obtained with channel noise, weargue the conclusion should be general for any zero-mean noise.The current SHH model is only an approximation to a much more complex reality. For example, it has presumed that thechannel dynamics are Markov chain process, they act independently, and the gating currents related to the movement of gatingcharges are negligible. However, those presumptions are not always tenable. For example, Schmid has shown that the gating3currents drastically reduce the spontaneous spiking rate if the membrane area is su ffi ciently large [37]. So our results should bereinvestigated with consideration of those factors. But we think those factors would not bring qualitative changes to our results,thus the general conclusions still hold.We then investigated the subthreshold signal detection in a neuronal network that concerns a coincidence detection neuron. Wefound by enhancing coincidence detection threshold and increasing the SHH neurons, the detection ability is greatly improved.It suggests that channel noise may play a role in information processing in the neural network level. In addition, correspondingto di ff erent coincidence detection thresholds, there exist an optimal number of neurons at which the total error is at its minimum.We have seen that this is also the result of balancing between responding to pulses and rejecting spontaneous firings. Since thisso-called double-system-size resonance phenomenon has been rarely reported [38], our work provides an example of such anobservation. In particular, with su ffi ciently large coincidence detection threshold, the total error is zero in a wide range of SHHneuron number, which means the detection ability of this network could be robust against variance of neuron number caused bycell production and death.We have shown that the reliable detection of subthreshold signals with the network is predictable with probability theory, aslong as each front layer neuron exhibits higher firing probability in response to signals than that induced by noise. Weak signaldetection is also important in practice. For example, mobile communications dictates the use of low power detection to prolongthe battery life. So our work suggests a possible way to design reliable stochastic resonance detector for weak signals [39]. Acknowledgements
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