Exhibit 99.1

For Immediate Release Wednesday, September 15, 2004

CYBERONICS REITERATES NO INTEREST IN ANY MERGER OR COMBINATION WITH
ADVANCED NEUROMODULATION SYSTEMS

Conference call at 4:30 PM EDT today, September 15

HOUSTON, Texas, September 15, 2004 — Cyberonics, Inc. (NASDAQ:CYBX) today announced that its Board of Directors, after consulting with its financial advisor, Merrill Lynch & Co., has unanimously rejected Advanced Neuromodulation Systems Inc.’s (NASDAQ:ANSI) invitation to “meet to discuss a business combination.” The Company reconfirms its previously stated position in its press release dated August 20, 2004, that it is not interested in any combination or merger and remains focused on strengthening its epilepsy business and gaining clarity and certainty in a revised depression regulatory plan and timeline.

CONFERENCE CALL ACCESS INFORMATION

A conference call to discuss this press release will be held at 4:30 PM EDT today, Wednesday, September 15, 2004. To listen to the conference call live by telephone dial 877-451-8943 (if dialing from within the U.S.) or 706-679-3062 (if dialing from outside the U.S.). The conference ID is 210644; the leader is Pam Westbrook. A replay of the conference call will be available two hours after the completion of the conference call on Wednesday, September 15, 2004 through Wednesday, September 29, 2004 by dialing 800-642-1687 (if dialing from within the U.S.) or 706-645-9291 (if dialing outside the U.S.). The replay conference ID access code is 210644.

ABOUT VNS THERAPY AND CYBERONICS

Cyberonics, Inc. (NASDAQ:CYBX) was founded in 1987 to design, develop and market medical devices for the long-term treatment of epilepsy and other chronic neurological disorders using a unique therapy, vagus nerve stimulation (VNS). Stimulation is delivered by the VNS Therapy System, an implantable generator similar to a cardiac pacemaker. The VNS Therapy System delivers preprogrammed intermittent mild electrical pulses to the vagus nerve 24 hours a day. The Company’s initial market is epilepsy, which is characterized by recurrent seizures. Epilepsy is the second most prevalent neurological disorder. The Cyberonics VNS Therapy System was approved by the FDA on July 16, 1997 for use as an adjunctive therapy in reducing the frequency of seizures in adults and adolescents over 12 years of age with partial onset seizures that are refractory to antiepileptic medications. The VNS Therapy System is also approved for sale as a treatment for epilepsy in all the member countries of the European Economic Area, Canada, Australia and other markets. To date, more than 27,000 epilepsy patients in 24 countries have accumulated over 79,000 patient years of experience using VNS Therapy. The VNS Therapy System is approved for sale in the European Economic Area and in Canada as a treatment for depression in patients with treatment-resistant or treatment-intolerant major depressive episodes including unipolar depression and bipolar disorder (manic depression). VNS Therapy is at various levels of investigational

clinical study as a potential treatment for depression, anxiety disorders, Alzheimer’s disease, and chronic headache/migraine. The Company is headquartered in Houston, Texas and has an office in Brussels, Belgium. For additional information please visit us at www.cyberonics.com.

CONTACT INFORMATION:

Pamela B. Westbrook	Helen Shik
Vice President of Finance and CFO	Vice President
Cyberonics, Inc.	Schwartz Communications
100 Cyberonics Blvd.	230 Third Avenue
Houston, TX 77058	Waltham, MA 02154
Main: (281) 228-7200	Main: (781) 684-0770 ext. 6587
Fax: (281) 218-9332	Fax: (781) 684-6500
pbw@cyberonics.com	hshik@schwartz-pr.com

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