Patent Publication Number: US-2010121407-A1

Title: Neuromodulation having non-linear dynamics

Description:
CROSS-REFERENCES TO RELATED APPLICATIONS 
     This application claims the benefit of U.S. Provisional Application No. 61/114,288 (filed on 13 Nov. 2008), which is incorporated by reference herein. 
    
    
     TECHNICAL FIELD 
     The present invention relates to the modulation of neural function and improving symptoms in patients suffering from certain medical conditions. 
     BACKGROUND 
     Electrical neuromodulation has been demonstrated to be useful for a variety of neurologic conditions. As such, attempts have been made to treat brain injury (e.g., due to trauma, hypoxia/anoxia, or stroke) by deep brain electrical stimulation. However, current approaches to neuromodulation for the treatment of brain injury and other conditions have had only limited efficacy. If electrical stimulation or other forms of neuromodulation are to have a greater impact on the treatment of brain injury (or other neurologic conditions), further improvements are needed. 
     SUMMARY 
     In one aspect, the present invention provides a method for neuromodulation in a live mammalian subject, comprising: applying electromagnetic energy to a site in the nervous system of the subject using a signal comprising a series of pulses, wherein the inter-pulse intervals are varied using the output of a deterministic, non-linear, dynamical system comprising one or more system control parameters. In certain embodiments, the electromagnetic energy is electrical. In certain embodiments, the dynamical system is a map ruled by a difference equation. In certain embodiments, the site in the nervous system is the brain. 
     In another aspect, the present invention provides a neuromodulation apparatus comprising: an electrode comprising an electrode contact; and a pulse generator coupled to the electrode; wherein the pulse generator is programmed to apply an electrical signal to the electrode contact, wherein the electrical signal comprises a series of pulses, wherein the inter-pulse intervals are varied using the output of a deterministic, non-linear, dynamical system having one or more system control parameters. In certain embodiments, the apparatus further comprises a physiologic sensor coupled to the pulse generator. 
     In another aspect, the present invention provides a computer-readable storage medium having executable instructions for performing the following: obtaining a set of solutions to one or more equations that rule a deterministic, non-linear, dynamical system having one or more system control parameters; and determining a set of inter-pulse intervals using the set of solutions, wherein the inter-pulse intervals define the time intervals between the pulses of a neuromodulation signal. In certain embodiments, the instructions further include controlling an apparatus to apply the neuromodulation signal to a site in the nervous system, such as the brain. 
     The present invention also provides methods of improving the symptoms in a patient suffering from certain medical disorders by applying electromagnetic energy to a site in the nervous system of the patient using a signal comprising a series of pulses, wherein the inter-pulse intervals are varied using the output of a deterministic, non-linear, dynamical system comprising one or more system control parameters. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIG. 1  shows a schematic illustration of a possible analogy comparing the brain&#39;s arousal function to different phases of matter. 
         FIG. 2A  shows a schematic illustration of ascending neuroanatomical pathways that may be involved in signaling arousal.  FIG. 2B  shows a schematic illustration of descending neuroanatomical pathways that may be involved in CNS arousal. 
         FIG. 3A  shows the solutions to a logistic map with control parameter R in a range of 2.4 to 4.0.  FIG. 3B  shows a plot of the output values of the logistic map for R=4.0 through 500 iterations. 
         FIG. 4  shows the solutions to a Hénon map for values of control parameter a in a range of 1.0 to 1.5 and b=0.3. 
         FIG. 5  shows a plot of the trajectory of the Lorenz system for σ=10, β=8/3, and ρ=28. 
         FIG. 6A  shows a neuromodulation apparatus according to an embodiment of the present invention.  FIG. 6B  shows a portion of the signal being applied by the neuromodulation apparatus. 
         FIG. 7  shows a flowchart of the operation of a neuromodulation apparatus according to an embodiment. 
         FIG. 8A  shows bar graphs of the measured activity for a mouse that was subjected to neuromodulation according to an embodiment of the invention.  FIG. 8B  shows the fixed-interval signal pattern used in the experiment.  FIG. 8C  shows the chaotic-interval signal pattern used in the experiment. 
         FIGS. 9A-9C  show bar graphs representing locomotor activity data obtained from mice in the arousal assay experiments.  FIG. 9A  shows the horizontal activity;  FIG. 9B  shows the total distance; and  FIG. 9C  shows the vertical activity. 
         FIGS. 10A and 10B  show bar graphs representing locomotor activity data obtained from mice in the telemetry-based experiments.  FIG. 10A  shows the results for the mice stimulated in the basal nucleus of Meynert; and  FIG. 10B  shows the results for the mice stimulated in the central-lateral thalamus. 
         FIGS. 11A and 11B  show the chaotic-interval signal patterns that were used in the experiments referred to in  FIGS. 10A and 10B .  FIG. 11A  shows Chaotic Pattern  1  and  FIG. 11B  shows Chaotic Pattern  2 . 
     
    
    
     DETAILED DESCRIPTION 
     The present invention relates to the modulation of neural function. In one aspect, the present invention provides a method for neuromodulation in a live mammalian subject, such as a human patient. The modulation of neural function can be useful in improving the symptoms of a patient suffering from a neurological disorder such as traumatic brain injury (TBI) or stroke. Particular deficits resulting from these conditions include, for example, language, motor and cognitive deficits and the methods of the present invention, in certain embodiments, are directed to improving such deficits in such patients. A method comprises applying electromagnetic energy to a target site in the nervous system, preferably the brain, of the subject using a signal comprising a series of pulses, wherein the inter-pulse intervals are varied using the output of a deterministic, non-linear, dynamical system comprising one or more system control parameters. 
     A dynamical system is a state space S (or phase space), a set of times T, and a rule R for evolution, that gives the consequent(s) to a state “s” (which is a member of S). The state space S has coordinates describing the state at any instant (“s”) and the dynamical rule R specifies the immediate future of all state variables, given only the present values of those same state variables. Thus, a dynamical system can be considered a model describing the temporal evolution of a state space according to a rule for time evolution. Dynamical systems are deterministic if there is a unique consequent to every state (as opposed to stochastic or random if there is a probability distribution of possible consequents). 
     The neuromodulation may be targeted to any of various sites in the nervous system of the subject. In certain embodiments, the target site may be a site in the brain involved in generalized CNS (central nervous system) arousal. Generalized arousal is a global CNS state that is believed to be a primitive driving force behind motivated behavioral responses, cognitive functions, and emotional expressions. Earlier neuroscience work in this area has focused on understanding the arousal system by its individual circuit components, e.g., how individual stimuli evoke specific motor responses. However, more recent neuroscience work has addressed how large classes of salient stimuli from multiple sensory modalities cause changes in the entire state of the brain. New models have been proposed to explain how arousal responses encompassing all sensory modalities drive a wide range of motor and emotional responses with extreme sensitivity to the initial state of the system, and with very rapid and highly reliable responses. It is believed by the inventor(s) that non-linear dynamics theory (e.g., chaotic dynamics) best explains this robustly complex arousal system in the brain, which changes through with time and is subject to multiple feedback loops. 
     Without intending to be bound by theory and for the purposes of illustration only,  FIG. 1  demonstrates a possible analogy that may be useful for understanding how a non-linear signal pattern may be effective in modulating brain or other neural function. This analogy compares the brain&#39;s arousal function to different phases of matter. The top portion of  FIG. 1  shows a schematic illustration of the molecular ordering for liquid crystals ranging from liquid phase at higher temperatures (towards disordered molecules on the right side) to crystalline phase at lower temperatures (towards well-ordered molecules on the left side). T denotes the temperature. Between the liquid phase and the crystalline phase is the liquid crystalline phase, which is highly-sensitive because of its proximity to a phase transition to the liquid phase. 
     By analogy, in an animal at rest, large numbers of arousal-related neurons have their firing rates subject to chaotic dynamics so that the effects of small perturbations from the arousing stimulus can be amplified selectively and very rapidly. When a movement in response to that stimulus is initiated, cortical and subcortical controls take over, moving the system across the nearby phase transition into the domain of orderly, high rates of firing. Thus, analogous to phase transitions in a liquid crystal, CNS arousal systems, having “woken up” the brain to activate behavior, go through a phase transition and emerge under the control of orderly movement control systems. Based on the dynamics of brain arousal systems that are set forth here, it is believed that brain arousal systems are sensitive to neuromodulation with inter-pulse intervals that vary according to the output of a non-linear process, such as a chaotic process. 
       FIGS. 2A and 2B  show the circuitry believed to be involved in CNS arousal mechanisms. As shown in  FIG. 2A , the classical neuroanatomical pathways ascending from the lower brainstem toward the forebrain can signal arousal using norepinephrine, dopamine, serotonin, histamine, and acetylcholine as transmitters. Four sensory modalities feed these ascending pathways: touch (including pain), taste, vestibular, and auditory. These ascending pathways include norepinephrine-containing systems (NE) that tend to emphasize projections to the more posterior cerebral cortex (P, except for occipital cortex) and to support sensory alertness. Dopaminergic systems (DA) tend to project more strongly to the anterior frontal cortex (A) and to foster directed motor acts. Serotonergic (5HT) neurons project preferentially to a more ancient form of cortex (“limbic cortex”) and hypothalamus, and to be involved in emotional behaviors and autonomic controls. Cholinergic neurons (ACh) in the basal forebrain support arousal by their widespread projections across the cerebral cortex. Histamine-producing neurons (HA) likewise have extremely widespread projections which actually originate in the hypothalamus and are strongly associated with increased CNS arousal. 
     Descending neuroanatomical pathways projecting from the forebrain toward the brainstem also play an important role in CNS arousal. As shown in  FIG. 2B , lateral hypothalamic area (LHA) orexin neurons project down to monoamine-expressing cell groups in the lower brainstem and even to the spinal cord. Histamine (HA)-containing hypothalamic neurons in the tuberomammillary nucleus (TMN) have widespread projections, and receive inputs from a ‘biological clock’, the suprachiasmatic nucleus (SCN). Preoptic area (POA) neurons have descending axons which affect sleep and autonomic physiology. For example, nerve cells in the preoptic area connect to lower brain regions, which control the viscera. Likewise, the paraventricular nucleus of the hypothalamus has axonal projections which, in principle, could contribute to all aspects of arousal: cerebral cortical, autonomic, endocrine and behavioral. Oxytocin (OT) and arginine vasopressin (AVP)-expressing neurons in the parvocellular portion of the paraventricular hypothalamic nucleus (PVNp) control autonomic arousal through the lower brainstem and spinal cord, and affect EEG arousal through projections to locus coeruleus. In sum, while the ascending arousal systems have relatively few neurons, only sparse abilities to encode particular stimuli and are responsible for ‘waking up’ the cerebral cortex, descending arousal systems prepare the body for action by empowering reticulospinal neurons to activate our big posture-supporting trunk muscles and by activating autonomic systems. 
     Based on this understanding of CNS arousal circuitry, exemplary targets for neuromodulation according to the present invention include sites in the central nervous system, including the brain and spinal cord, and the peripheral nervous system, including spinal and autonomic nerves. Certain deep brain sites that could be targets include the thalamus (e.g., central, anterior, posterior, or intralaminar portions such as the intralaminar nuclei), basal forebrain (e.g., basal nucleus of Meynert), hypothalamus (e.g., anterior hypothalamic nucleus, tuberomammillary nucleus, suprachiasmatic nucleus, preoptic area, paraventricular nucleus, etc.), or the brainstem (e.g., locus coeruleus, mesencephalic reticular formation, laterodorsal tegmentum (LDT) nuclei, pedunculopontine tegmentum (PPT) nuclei, etc.). 
     The inter-pulse intervals in the neuromodulation are varied using the output of a deterministic, non-linear, dynamical system comprising one or more system control parameters. Various types of deterministic, non-linear, dynamical systems are known in the art and are suitable for use in varying the inter-pulse intervals. For example, the dynamical system may be a discrete-time dynamical system that outputs a sequence S n  at discrete times n=[1 . . . . N]. A deterministic evolution rule with discrete time and a continuous state space is called a “map” and its evolution is defined by the iteration: 
         s   t+1   =f ( s   t ) 
     In some cases, the dynamical system may be capable of exhibiting chaotic behavior. One particular example of a time-discrete dynamical system capable of exhibiting chaotic behavior is the logistic map produced by the following difference equation: 
         x   (n+1)   =Rx   n (1 −x   n ) 
     wherein the constant R is a system control parameter having a value between 0 and 4 (inclusive), and each X n  is between 0 and 1, with an initial value being chosen to begin the iterative process.  FIG. 3  shows the solutions to this logistic map and demonstrates that the logistic map exhibits either simple or complex behavior depending on the system control parameter R. The solutions become chaotic with R=3.5 and it is known that approximately 90% of R values between 3.57 and 4.0 results in chaotic behavior.  FIG. 3B  shows the output values X n  of the logistic map for R=4.0 through 500 iterations. 
     Another example of a time-discrete dynamical system capable of exhibiting chaotic behavior example is a Hénon map produced by the following coupled difference equations: 
         x   n+1 =1−α x   2   n   +y   n    
     
       
      
       y 
       n+1 
       =βx 
       n  
      
     
     This map depends on two system control parameters, α and β.  FIG. 4  shows the solutions to the Hénon map for β=0.3, demonstrating that the map exhibits chaotic behavior for various values of α. For example, the Hénon map is known to exhibit chaotic behavior at α=1.4 and β=0.3. 
     Another example of a time-discrete dynamical system capable of exhibiting chaotic behavior is the Standard map defined by the difference equations: 
         p   n+1   =p   n   +K  sin(θ n ) 
       θ n+1 =θ n   +p   n+1    
     where angular positions p n  and θ n  are taken modulo 2π and the constant K is the system control parameter having a value &gt;0. 
     Other dynamical systems are on a manifold that is continuously differentiable with respect to time (such a dynamical system is called a “flow”). One such dynamical system is the Lorenz system governed by the following differential equations: 
     
       
         
           
             
               
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     where x, y, and z are the state variables; t is the independent variable; and the constants α, ρ, and β are the system control parameters having a value &gt;0. The Lorenz system is known to exhibit chaotic behavior for σ=10, β=8/3, and ρ=28 and the trajectory plot is shown in  FIG. 5 . 
     As demonstrated above, the system control parameter(s) of the dynamical system can be selected such that the dynamical system exhibits chaotic behavior. The term “chaotic behavior” means that the system exhibits long-term aperiodic behavior with a sensitivity to initial conditions, i.e., the fact that any two trajectories of the system, no matter how closely their initial starting positions are, will eventually diverge, and such divergence will be of exponential order. One measure of divergence of trajectories in a dynamical system is the Lyapunov exponent, which is a measure of the average rate of divergence/convergence of nearby trajectories. This can be used to determine whether the system is periodic, chaotic, or at equilibrium. 
     The Lyapunov exponent provides such a measure by comparing a reference orbit with a displaced orbit. Iterates of the initial condition x 0  are denoted the reference orbit and the displaced orbit is given by iterates of the initial condition x 0  where d 0  is a vector of infinitely small length denoting the displacement from the initial condition x 0 . The initial orientation of the initial displacement is given by u 0 =d 0 /|d 0 |. Using this notation, one way of calculating the Lyapunov exponent is as follows: 
     
       
         
           
             
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     where d n , is the deviation of the displaced orbit from the reference orbit, given by the n&#39;th iterate of x 0 . A positive Lyapunov exponent indicates a chaotic state. 
     The neuromodulation may be implemented using any type of electromagnetic energy suitable for modulating neural tissue and in addition or alternatively, any form of energy suitable for modulating neural tissue. Such suitable types of electromagnetic energy include, for example, electrical, optical, magnetic, or radiofrequency (RF) energy. Alternatively, ultrasound energy could be used to implement the neuromodulation. 
     In certain embodiments, the present invention is implemented using electrical energy. The electrode may be any of those known in the art that are suitable for use in neuromodulation. The design characteristics of the electrode will vary depending upon the needs of the particular application, including such features as the number, direction, position, and/or arrangement of electrode contacts on the electrode; number of independent channels; and geometry and/or configuration of the electrode. 
     The electrical energy being applied may be characterized according to various parameters, including voltage, current amplitude, pulse width, average pulse frequency, train length, or waveform. Such signal parameters will vary depending upon the particular application. For example, the voltage may be selected from a range of ±0.1-10 V, pulse width may be selected from a range of 50-500 μs per phase, average pulse frequency may be selected from a range of 30-300 Hz, and current may be selected from a range of ±0.1 μA-5 mA. The electrical signal can have any suitable waveform, including square, sinusoidal, sawtooth, spiked, exponential rise/decay, or Gaussian, and where applicable, the signal may be monophasic, biphasic, multiphasic, or asymmetric. In some cases, the average pulse frequency is 200 Hz or greater. 
     Referring to the example embodiment shown in  FIGS. 6A and 6B , a neuromodulation apparatus  30  includes an electrode  32  having electrode contacts  34 , which is implanted in a brain site  40 . A lead extension  38 , which travels in a subcutaneous tunnel created by blunt dissection, connects electrode contacts  32  to a pulse generator  50  implanted in a subcutaneous pocket in the patient&#39;s chest area. As such, electrode contacts  34  are coupled to pulse generator  50 . As used herein, the term “coupled” refers to a signaling relationship between the components in question, including direct connection or contact (e.g., via an electrically or optically conductive path), radio frequency (RF), infrared (IR), capacitive coupling, and inductive coupling to name a few. 
     Pulse generator  50  is programmed to generate an electrical signal based on outputted solutions x n  to the logistic map above. The logistic map may be solved before or during the neuromodulation process. For example, solutions to the logistic map may be solved in advance and stored for later retrieval, or alternatively, the solutions may be calculated while the neuromodulation is in progress (e.g., in real-time). The output from the logistic map may be applied in any suitable manner to set the inter-pulse intervals. In some cases, the inter-pulse intervals may be some function of the output solutions (x n ). For example, output solutions may be scaled in an appropriate manner taking into consideration various factors such as the performance limitations of the neuromodulation equipment, the desired average pulse frequency, and the desired number of pulses in the train. 
     In operation, pulse generator  50  generates a signal and transmits the signal via lead extension  38  to electrode contacts  34  on electrode  32 .  FIG. 6B  shows a schematic representation of the signal being applied at electrode contacts  34 . The signal is a series of biphasic voltage pulses separated by time intervals Lx n . Each x n  is the n&#39;th iterated output of the logistic equation with R set to a value between 3.57 and 4.0 such that the logistic equation generates a chaotic output. The output of the logistic equation for n=[90 . . . 94] are arbitrarily selected for representation here. Each L is a constant used as a scalar multiplier to convert each x n  to a time interval that is scaled to produce an electrical signal having a desired average pulse frequency (e.g., 50 Hz). Thus, the inter-pulse intervals in the signal shown in  FIG. 6B  are Lx 90 , Lx 91 , Lx 92 , Lx 93 , and so on. 
     The various functions and capabilities of neuromodulation apparatus  30  may be performed by electronic hardware, computer software (or firmware), or a combination of both. As such, neuromodulation apparatus  30  may include a computer-readable storage medium having executable instructions for performing the various processes as described and illustrated herein. The storage medium may be any type of computer-readable medium (i.e., one capable of being read by a computer), such as hard drive memory, flash memory, floppy disk memory, or optically-encoded memory (e.g., a compact disk, DVD-ROM, DVD±R, CD-ROM, CD±R). The systems disclosed herein may also include addressable memory (e.g., random access memory or cache memory) to store data and/or sets of instructions that may be included within, or be generated by, the executable instructions when they are executed by a processor on the respective platform. For example, pulse generator  50  may have executable instructions for performing the calculations needed to produce the desired neuromodulation signal.  FIG. 7  shows a flowchart of how a neuromodulation apparatus may be operated according to an embodiment of the present invention. 
     In certain embodiments, the pulse generator is pre-programmed to deliver an electrical signal of a predetermined pattern to modulate neural function as described below or to treat neural conditions or disorders (i.e., to improve symptoms) as described below. In a preferred embodiment, the pulse generator is pre-programmed to deliver an electrical signal of a predetermined pattern to improve the function(s) (cognitive, motor, psychiatric, or other deficient functions) of a patient suffering from stroke or traumatic brain injury. 
     In certain embodiments, the present invention further comprises modifying the neuromodulation signal based on feedback data obtained from the subject. The feedback data may be any condition of the subject that is useful in measuring the effectiveness of the neuromodulation. For example, neuromodulation apparatus  60  above may have a sensor for detecting or measuring a physiologic parameter such as mechanical, motion, electrical, and/or chemical activity on or within the subject&#39;s body. Such physiologic parameters may be detected in various parts or functions of the body, including the nervous system, endocrine system, musculoskeletal system, respiratory system, circulatory system, urinary system, and/or digestive system. Examples of electrical activity that could be monitored include neuronal electrical activity, such as the electrophysiologic signals from the brain (e.g., EEG or electrode recordings), or muscular electrical activity (e.g., EMG). Examples of chemical activity that could be monitored include the detection or measurement of neurotransmitters, hormones, neuropeptides, or electrolytes in the subject&#39;s body (e.g., in the brain, blood, or cerebrospinal fluid). Other examples of physiologic parameters include heart rate, respiratory rate, blood pressure, blood oxygenation, etc. Sensors could also be used to detect motion or movement (e.g., for motor activity, tremors, gait, etc.). 
     In some cases, the feedback data may be indicative of the generalized arousal state of the subject. According to one proposed definition, generalized arousal has three components: (1) alertness to sensory stimuli in any one or more sensory modalities; (2) voluntary motor activity; and (3) emotional reactivity. All three components can be measured objectively by changes in physical activity. There are also various ways to quantify the arousal state. For example, according to one proposed mathematical model, arousal is a compound function of its principal components as follows: 
         A=F   g ( A   g )·[ F   1 ( A   C1 )+ F   2 ( A   C2 ) . . . + . . .  F   n ( A   Cn )] 
     where A is the state of global CNS arousal, A g  is generalized arousal, each A Cn  is a specific form of arousal (e.g., sexual, hunger, thirst, salt hunger, fear, and pain), and each F n  is the relative force of that arousal component. 
     Feedback algorithms for modifying the neuromodulation signal according to the feedback data may increase or decrease the amount of arousal, depending upon the particular application. For example, the feedback algorithm may change the system control parameters of the dynamical system (e.g., “walking through” a series of R values for the logistic map), change the set of sequence terms used to vary the inter-pulse intervals, or change the number of sequence terms in a repeated-set used to vary the inter-pulse intervals. 
     The present invention can be used for neuromodulating a site in the nervous system of a live mammalian subject. Such neuromodulation includes activating or inhibiting neural tissue and includes modulating neural functions such as stimulating, depressing, or enhancing neural function (abnormal or normal) or treating neural conditions and disorders (i.e., to improve symptoms). 
     In preferred embodiments, the neurologic disorders are stroke or traumatic brain injury (and the symptoms of such disorders are improved, for example, by neuromodulation of the thalamus, such as the intralaminar nuclei of the thalamus). In certain preferred embodiments, the methods of the present invention are used to improve cognitive, psychiatric, motor, and/or other functions in patients suffering from stroke and/or traumatic brain injury. In some embodiments, the neurologic disorders or conditions treated (i.e., to improve the symptoms) by the present invention are characterized by arousal dysfunction. Such neurologic disorders or conditions that involve arousal dysfunction include, for example, coma, stupor, and sleep disorders. Non-limiting examples of sleep disorders include hypersomnia, insomnia, and narcolepsy. Other neurologic disorders include disorders of attention or mood such as, for example, depression, bipolar disorder, distractibility, inattention, locked-on vigilance, obsessiveness, and attention deficit hyperactivity disorder; disorders of affect or emotion such as, for example, anxiety or panic attacks, agitation, irritability, lack of restraint, logorrhea, aggression, apathy, akinesia, mutism, autism, dyslexia; disorders of psychic energy such as, for example, indifference, chronic fatigue syndrome, fibromyalgia, and chronic pain (including neuropathic pain); disorders of global cognitive function such as, for example, delirium, fugue states, dementia (e.g. age-related, Alzheimer&#39;s, multimodal, etc.), and vegetative state; impairments of focal conscious properties such as agnosia, apraxia, aphasia, loss of anticipation, and amnesia; and brain injury (e.g., due to trauma, stroke, infection, etc.). 
     Experimental 
     Experimental trials were conducted in which mice were subjected to neuromodulation according to certain embodiments of the present invention. Electrodes were surgically implanted into the brains of the mice for deep brain stimulation. For the arousal assay experiments, the mice were individually housed inside an acrylic cage (i.e., arousal assay box) of a VersaMax animal monitoring system (AccuScan Instruments Inc., Columbus, Ohio). The cages were equipped with horizontal and vertical sensors containing a set of infrared photo beams distributed side-to-side and front-to-back. A VersaMax Analyzer (AccuScan Instruments Inc., Columbus, Ohio) was used to collect the beam status information from the arousal assay box. Each disruption of a beam was recorded as an activity count. 
     Locomotor activity was measured according to three main parameters: (a) horizontal activity (HACTV)=the total number of beam interruptions in the horizontal sensor within the observation time-frame; (b) total distance (TOTDIST)=the distance traveled around the entire cage in a continuous path, in cm, within the observation time-frame; and (c) vertical activity (VACTV)=the total number of beam interruptions of the vertical sensors within the observation time-frame. 
     To acclimate the mice, the mice were handled and plugged in without stimulation once a day for 3 or 4 days before stimulation began. On stimulation days, mice were handled using the same protocol and stimulated for 10 minutes before being returned to the arousal box. The mice were then subjected to at least one cycle of fixed-interval stimulation followed by chaotic-interval stimulation (e.g., fixed-interval on day  1 , followed by chaotic-interval on day  2 ). The fixed-interval stimulations were provided at a frequency of 50 hz and the chaotic-interval stimulation was provided at an average frequency of 50 hz. The pulses were of 0.1 msec duration. 
     To generate the series of chaotic intervals, the output from the logistic equation was generated to a thousand or more terms. From this set of a thousand or more terms, a subset of contiguous terms were selected for use. For example, the set of terms may be the last 10, 15, or 50 output terms in the sequence generated by the logistic equation. Next, the maximum operable frequency f (based on instrument limitations) was determined and this was converted into the minimum operable inter-pulse intervals (IPI) using the relation 1/f=IPI. Next, a multiplier k was defined that sets the output terms to the minimum operable inter-pulse interval (IPI). Next, based on the number of pulses j desired in the train (e.g., j=10) and the multiplier k, a consecutive series of IPI&#39;s (P 1 , P 2 , P 3 , . . . Pj) that would produce a desired average frequency AF (e.g., 50 Hz) was calculated using the following formula: 
     
       
         
           
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             = 
             
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                 · 
                 
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                       1 
                     
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                       3 
                     
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                     … 
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     Another set of experiments were performed on mice using a telemetry system (Data Sciences International, St. Paul, Minn.), which included signal receivers, signal transmitters, and data collection/analysis software. Electrodes were implanted in the brain of the mice and connected to the telemetry transmitter/receiver system. The system includes an activity sensor, as well as two channels for biopotentials, one for electroencephalogram (EEG) and one for electromyogram (EMG). As such, in addition to activity data, EEG (electroencephalography) and EMG (electromyogram) data were also collected and analyzed. 
     Before the start of each study, individually-housed implanted mice were placed on telemetry receivers within a grounded faraday cage. Transmitters were turned on right before recording started. The day after the start of recording, mice were stimulated 4 times a day. Recordings were stopped either 2 hours after the last stimulation or the next morning. 
     Within a few days after the last stimulation, the mice were perfused and the brains removed. After post-fixing and dehydration with 30% sucrose, the brains were sliced at 60 μm thickness, mounted on slides, and Nissl stained with cresyl violet dye to confirm placement of electrodes. 
       FIGS. 8A-8C  show the results of one of the mice in the arousal assay experiments in which the stimulation electrodes were implanted bilaterally in the basal (B) nucleus of Meynert.  FIG. 8A  show bar graphs of the activity data during the observation time-frame. After acclimation, the mouse was handled and activity recorded before and/or after manipulation. A sham stimulation (no stimulation) trial was performed with activity recorded 30 minutes before and/or after. Subsequently, the mouse was subjected to a fixed-frequency stimulation for 10 minutes at 130 Hz using biphasic square wave pulses, with activity recorded 30 minutes before and/or after.  FIG. 8B  depicts the pulse pattern used in the fixed-frequency stimulation. The mouse was then subjected to a chaotic pulse train stimulation for 10 minutes at an average frequency of 50 Hz using biphasic square wave pulses.  FIG. 8C  depicts the pulse pattern used in the chaotic stimulation, which was a series of 15 pulses spanning 0.3 seconds that were repeated for 10 minutes. As seen in  FIG. 8A , the effectiveness of chaotic stimulation was dramatically better than fixed-frequency stimulation. 
       FIGS. 9A-9C  show bar graphs of data (obtained from at least 5 mice) from the arousal assay experiments. The bar graphs represent the difference in activity measured before and after stimulation.  FIG. 9A  shows the horizontal activity;  FIG. 9B  shows the total distance traveled; and  FIG. 9C  shows the vertical activity. In each case, the chaotic pulse train stimulation had a different effect on activity as compared to the fixed-frequency stimulation. For horizontal and vertical activity, the chaotic stimulation resulted in a greater increase in activity than fixed-frequency stimulation. For total distance, the chaotic stimulation resulted in a smaller increase in activity than the fixed-frequency stimulation. 
       FIGS. 10A and 10B  show bar graphs of data (obtained from at least 5 mice) from the telemetry-based experiments. The bar graphs represent the difference in activity measured before and after stimulation.  FIG. 10A  shows the activity results for fixed-frequency stimulation and two different chaotic train stimulations for mice with electrodes implanted in the basal (B) nucleus of Meynert.  FIG. 10B  shows the activity results for fixed-frequency stimulation and the two different chaotic train stimulations for mice with electrodes implanted in the central-lateral thalamus. As seen in  FIGS. 10A and 10B , one of the chaotic pulse trains (“Chaotic  1 ”) was substantially more effective than the fixed-frequency stimulation, while the other (“Chaotic  2 ”) was not. 
       FIG. 11A  depicts the pulse pattern used in Chaotic  1  of  FIGS. 10A and 10B , and  FIG. 11B  depicts the pulse pattern used in Chaotic  2 . Chaotic Pattern  1  was a series of 10 pulses of 200 msec width each that was repeated for 10 minutes at 50 Hz average frequency. Chaotic Pattern  2  was a series of 50 pulses of 200 msec width each that was repeated for 10 minutes at 50 Hz average frequency. 
     In certain embodiments, a finite set of contiguous terms outputted by the dynamical system is selected and this finite set of terms is used to generate a repeating pattern for the neuromodulation signal. As seen in the above experiments, using a set of 10 and 15 contiguous terms outputted by the logistic equation produced improved results over fixed-frequency stimulation. However, using a set of 50 contiguous terms (Chaotic Pattern  2 ) outputted by the logistic equation did not produce improved results. Based on these results, it may be desirable to generate signal patterns that are close to the transition between non-linear dynamics and orderly patterns. Thus, in some cases, the set of contiguous terms selected from the output of the dynamical system is less than 50 contiguous terms; and in some cases, in the range of 5-45 contiguous terms. 
     The foregoing description and examples have been set forth merely to illustrate the invention and are not intended as being limiting. Each of the disclosed aspects and embodiments of the present invention may be considered individually or in combination with other aspects, embodiments, and variations of the invention. Further, while certain features of embodiments of the present invention may be shown in only certain figures, such features can be incorporated into other embodiments shown in other figures while remaining within the scope of the present invention. In addition, unless otherwise specified, none of the steps of the methods of the present invention are confined to any particular order of performance. Modifications of the disclosed embodiments incorporating the spirit and substance of the invention may occur to persons skilled in the art and such modifications are within the scope of the present invention. Furthermore, all references cited herein are incorporated by reference in their entirety.